What is the difference between translation and rotation? What is the difference between translation and rotation ?
If this were a mathematics site, the question would be at best naive.
But this is physics site, and the question must be interpreted as a
question about physical theory, that is about hypotheses that can be tested,
subjected to experiments and possibly disproved by experiments.
Restatement of the question
After about 3 days, 5 answers, 160 views and some more comments,
taking these contributions into account (hence the length).
First I thank all users who commented or tried to answer my ill
stated question, and I apologize for not doing better. Hopefully, they
did help me improve my understanding and the statement of my
question. You can look at the awkward prior formulations of the
question which better explain the existing answers.
I am trying to understand whether and how translation differs from
rotation, and whether or why it is a necessary physical concept or
possibly only a mathematical convenience.
There are two sides to the issue I am raising, one regarding
space(time) symmetries, and one regarding motion. From what (very)
little I understand of Noether's theorem, these cannot be unrelated as
laws governing motion have to conserve charges that are derived from
the space(time) symmetries. This may have been one source of my
initial confusion.
One of my point is that if there are situations when rotations is not
distinguishable from translation: infinitesimal angles of rotation as
suggested by user namehere. Then relevant phenomena can be
analyzed either as rotations or as translation, with proper
accomodation, particularly to account for the existence of a radius
when rotation is concerned, which changes dimensionality and the
mathematical apparatus.
Of course this requires "care" when considering phenomena involving the
center of rotation or phenomena indefinitely distant.
One example is torque, moment of inertia, angular momentum, vs force,
mass and momentum. The possible undistinguishability of translation and
rotation would seem to indicate that they are really two guises for
the same set of phenomena. They relate to two distinct symmetries, but
is that enough to assert that they are fundamentally different ? This
is precisely what is bothering me in the last comments of user
namehere attached to his answer and motivated my question initially.
Actually, it was someone telling me that "angular momentum is not
linear momentum going round in a circle" that started me on this
issue, as I was not convinced. It may have other manifestations, but it is also that.
I am aware that the mathematical expressions, including
dimensionality, are significantly different for translational and
rotational concepts, and somewhat more complex for rotational
concepts, as remarked by user namehere. But rotational concepts have
to account for the existence of a center and a radius which may be
directly involved in the phenomena being considered: this is typically
the case for the moment of inertia which has to account for a body
rotating on itself.
If we consider a translational phenomenon about force, mass and
momentum, occuring in a plane. We can analyse it indifferently as
translational, or as rotational with respect to a center of rotation
sufficiently distant on a line orthogonal to the plane, so that all
radiuses may be considered vectorially equal up to whatever precision
you wish. Since the radiuses may be considered equal, they can be
factored out of the rotational formulae to get the translational ones.
That is, the rotational mathematics can be approximated arbitrarily
well by its translational counterpart. This should accredit the
hypothesis that it is the same phenomena being accounted for in both
cases.
I am not trying to assert that rotating frames should be inertial
frames. I am only asking to what extent physicists can see a
difference, and, possibly (see below), whether inertial frames
actually exist. When do rotational phenomena differ in substance from
translational ones ? Is there some essential phenomenon that is
explained by one and not by the other ? And conversely ?
Mathematics is only a scaffolding for understanding problems. They are
not understanding by themselves. Mathematical differences in
expression do not necessarily imply a difference in physical essence.
Then one could also question whether translation is (or has to be) a
meaningful physical concept. This can be taken from the point of view
of space(time) symmetries or from the point of view of motion.  Why
should it be needed as a physical concept, or can it (should it) be
simply viewed as a mathematical convenience ? Does it have meaning
independently of the shape (curvature) of space ? (I guess relativists
have answers to that).
Take the 2D example of the surface of a sphere. What is translation in
that space ? The usual answer is "displacement along a great
circle".  This works for a point, but moderately well for a 2D solid,
as only a line in that solid will be able to move on a great circle.
I guess we can ignore that, as any solid will be "infinitesimal" with
respect to the kind of curvature radius to be considered.
However there is the other problem that, very simply,
every translation is a rotation, and in two different ways, with a
very large but finite radius. But it probably does not matter for
scale reasons.
Now, there is also the possibility that I completely missed or
misunderstood an essential point. Which would it be ?
 A: I hope I am interpreting your question right: One of the defining differences between translation and rotation is that translation is commutative. If I move forward 1 and right 1, it is the same as moving right 1 and then forward 1. The same can not be said of rotation. If I rotate my phone clockwise parallel to my body then clockwise perpendicular to my body I end up with a different end position than if I do the same rotations in a different order. 
We don't measure translation or rotation though. We can define a translation by measuring distance, speed, direction, etc. We can define a rotation by measuring angles, angular velocity, direction, etc. These measurements are limited by the precision of our measuring equipment. 
Any real motion is a combination of many different motions. Pure translation is really how we simplify real motion by excluding the things we don't care about. It is a simplification, and as such, I'm not sure it needs to be "proven".
A: If I am understanding your question correctly, you are asking how to define a inertial frame. This question can be considered in three different levels. 
In Nowton's theory, space and time are absolute. All motions are defined as relatvie motion in the obsolute space and time. A inertial frame is defined as a frame in which Newton's second law holds. This absolute motion is not observable because ethe (the absolute space frame) does not exist. 
In special relativity (without gravity), all motions are relative because space and time are relative. All inertial frams(where there is no gravity) are equilvalent. So the motions (translation or rotaion) can be observed as long as you set up a coordinate system in your frame.  The obsovation of motions becomes a `mathematical' problem as babou said at the beginning.  
In general relativity, since gravity exists, inertial frames can only be defined locally. If the accelerations cancels the gravity (owing to the equivalence preinciple, this can always be done locally) in a frame, then this frame can be considered as a inertial frame where special relativity works. 
Also, in the same way, what is rotation ? If there is a physical significance to translation and rotation, how precisely can they be physically distinguished. For example, what will distinguish a translation from a rotation with a very remote center ?

I think the question will be much clearer if one considers concrete objects.
If the object is point-like, rotation by itself has no meaning(if we don't consider spin). In that sense,  we cannot 'distinguish a translation from a rotation with a very remote center'.
But if the object has a finite size, rotation and translation are very different, because we have to set coordinate systems in the object frame to correctly describe the motion of every point. 
A translation may or may not be observed, depending on concerete models. For example, an infinite crystal is invariant under translation of one lattice constant. This translation is not observable since the system goes back to itself. If we translate the crystal by half lattice constant, the system does not go back and the translatoin is obervable. 
The translational invariance has physical consequences. If the translational invariance (or the periodicity) of the crystal is slightly violated by small perturbations, phone[?] excitations will be excited in the crystal which can be observed.
Similar discussion holds for rotation. 
A: "What will distinguish a translation from a rotation with a very remote center?"
There is no difference, at least in a limit. Indeed, infinitesimal rotations share many properties with translations, such as being commutative, unlike finite rotations, and in a sense all translations can be considered as rotations.
I believe however you are trying to ask about the significance of these translations and rotations in physics. I will try to answer what I think you are trying to ask.
The significance of translation and rotation in physics is mainly NOT as forms of motion, but as basic symmetries of space(time). In most physical theories, space is considered to be the same under arbitrary space(/time) translations and rotations about any spacial point. The significance of rotations and translations are precisely the fact that they do not alter the spacial structure or indeed the laws of physics in any single way. The symmetry offered by space towards these operations are what gives translations and rotations significance in physics.
I belive this makes clear to you the significance of these operations in physics.
A: partial answer
Symmetries as composition of other symmetries
Rephrasing my question, and better understanding it (thanks to all
contributions) lead me to what I consider at least a partial answer,
which is the following:
"Symmetry with respect to all rotations about all points implies
translational symmetry with respect to all translations" (from
Wikipedia), which I should have thought of earlier.
I guess that implies that conservation of momentum is somehow a
special case of conservation of angular momentum. Hence any phenomenon
that is explained by conservation of momentum should be explainable by
conservation of angular momentum.
But what about the reverse.
Another point concerns space with positive curvature. On a sphere,
symmetry with respect to all translations implies symmetry with
respect to all rotations, since composition of 2 translations can
change arbitrarily the orientation of a figure. However it is not
clear that this would have any effect on the issues considered here.
The reason is that it requires non local translations, which would have
no physical sense at cosmological scale, and it does not work using
only infinitesimal translations to get arbitrary infinitesimal
rotations.
Hence the above quote from Wikipedia is not enough. It is correct to
consider that translational symmetry is implied by rotational symmetry
only because it is also true when considering only infinitesimal
symmetries.
A: I think the concept that you are dealing with is the difference between Euclidean geometry and projective.  In Euclidean geometry you have parallel lines, and translations are the orientation and metric preserving transformations that take parallel lines to other parallel lines.  All other orientation and metric preserving transformations are rotations, which take lines into non parallel lines.
Both rotations and translations can be built out of a reflection in a line.  Two reflections about parallel lines generates a translation, two reflections about intersecting lines generates a rotation.  These can also be categorized by the number of fixed points.  Translations have zero, rotations have 1 and reflections have two (actually a whole line full).
In projective geometry, there are no parallel lines so the distinction between translations and rotations goes away.  They are both the same kind of transformation, the only distinction being for a particular embedding of Euclidean geometry inside Projective geometry (Ie. you have to consider the line at infinity a "special" line).
A: Rotation and Translation
The distinction is crucial for understanding Newton’s mechanics.
In broad terms, translation indicates motion in a straight line; and rotation indicates motion around an axis.
When Newton muses in Principia that maybe there is no such thing as a body truly at rest, he is also implying that maybe there is no such thing as motion in a straight line; i.e.  that all motion is ultimately curvilinear.
As usual (especially for Newton), the distinction traces back to Aristotle, and specifically to his discussion of ‘quantity’: “Quantity” means that which is divisible into constituent parts, each or every one of which is by nature some one individual thing. (Metaphysics, Bk V, xiii 1-5).
A quantity, then, has to be divisible (i.e. able to be divided); and it has be be divisible into parts that together constitute the whole; and each and every part must be some one individual thing by its very nature.
(The issue goes back to Zeno’s Paradox and how can a line be divided.)
Aristotle then identifies 2 kinds of quantity, each of which indicate different sorts of parts: 
(i) plurality - a numerically calculable quantity,
        potentially divisible into non-continuous parts;
and (ii) magnitude - a measurable quantity,
        potentially divisible into continuous parts.
Aristotle gives length and breadth and depth as kinds of magnitude or measurable quantity, since they involve continuity in direction (i.e. continuity in 1 direction, 2 directions, or 3 directions).  At the limits of direction of each kind (i.e. when each is considered as a single unit), such magnitudes become numerable as plurality of quantity and thereby present as a line (the delimited - and therefore numerable - unity of length) or a plane (as the delimited - and therefore numerable - unity of breadth) or a body (as the delimited - and therefore numerable - unity of depth).
‘Rotation’ applies to ‘magnitude’;
        measurable quantity of continuous parts 
        (think of fluids, like 5 pints of water, for example).
‘Translation’ applies to ‘plurality’;
        numerable quantity of non-continuous parts
        (think of 5 fish, for example).
Newton introduces ‘translation’ into his discussion of motion at the beginning of Principia:
Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another.
Note that the description depends upon Newton’s understanding of ‘place’ (as distinct from ‘space’) and is aristotelian.  The literal latin meaning of ‘translation’ (Newton is writing in latin) entails the acts of picking up and plonking down; a discontinuity is involved.  Such is the progression of number, from 2 to 3 to 4 etc.
Newton’s usage has to be understood within the context of the complex number plane, since ‘translation’ applies to the ‘real’ co-ordinates of the complex number plane.  However, ‘translation’ originates with the ‘imaginary’ co-ordinates; where rotation applies.
Rotation in the complex number plane-body involves  fourfold anti-clockwise dynamic motion 
    from -i to i to -1 to +1.
The transition from +1 to 1squared (which is the proper beginning of the linear progression of natural number) requires a discontinuity and another full dynamic sequence of  -i to i to -1 to +1.  Such is the basis for all clocks, numerating (by means of translation - picking up and plonking down) that dynamic sequence of  -i to i to -1 to +1.
Newton defines ‘number’ as a ratio of quantities:
By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity of the same kind, which we take for unity.
(Universal Arithmetic: Or, a Treatise of Arithmetical Composition and Resolution, in D.T. Whiteside(ed), The Mathematical Works of Isaac Newton, Vol. 2, New York(Johnson Reprint C)1967 pp.3-134.)
Hence the natural numbers are to be conceived dynamically, as squares, such that the series is properly 1sq'd/1, 2sq'd/2, 3sq'd/3 ...
Rotational motion in the complex number plane-body is the newtonian basis for the notion of spin in quantum physics.  It is the basis upon which the atomic nucleus is structured.  Moreover, it is the newtonian basis for the theoretical understanding of quantum fluctuations.
Newton emphasises that the principles of his physics (his natural philosophy) are mathematical.  He emphasises that space and place and motion and time are mathematical quantities.  Newton is not usually taken at his word in this, and his commentators are satisfied with applying mathematical symbols and formulae to data about space and place and motion and time; ignoring his insistence that space and place and motion and time are themselves mathematical quantities.
Clarification of the distinction between rotation and translation helps to reinvigorate Newton’s legacy and assert its relevance for an understanding of the problem areas in modern particle physics.
A: Translations and rotations are not different. A rotation is a general state of motion, and a pure translation is a degenerate form of rotation. In fact a translation is just an indication that rotation occurs at a distance, just as a torque is an indication that a force is acting on a distance and angular momentum is an indication that something at a distance has momentum.
See (the first part) of this answer on how linear and angular quantities work together to describe motion, loading or momentum.
An although this question is about forces and moments, in my answer I show that linear and angular motion are really manifestation of the same thing (the screw motion of a body) and they work together to fully describe a system.
So in mechanics there are three primary vectors that have a direction and magnitude only:


*

*Angular Velocity,  $\vec{\omega}$

*Force, $\vec{F}$

*Momentum, $\vec{p}$


and there are three secondary vectors that are moment vectors of the first which describe where the primary vectors occur:


*

*Linear Velocity, $\vec{v} = \vec{r} \times \vec{\omega}$

*Torque, $\vec{\tau} = \vec{r} \times \vec{F}$

*Angular Momentum, $\vec{L} = \vec{r} \times \vec{p}$


Here $\vec{r}$ is always defined from the point of measurement (coordinate origin) to the location of the action.
The general rule for where a primary vector acts is $$ \vec{r} = \frac{\text{(primary vector)} \times \text{(secondary vector)}}{\| \text{(primary vector)} \|^2}$$
You can see proof of this in the linked answers.
So is translation different from rotation? No, they are both manifestations of the same thing.
How? 


*

*To construct a rotating motion from the rotation axis direction $\vec{e}$ and location $\vec{r}$ with speed $\omega$ do this:


$$ \begin{align} \vec{\omega} & = \omega \vec{e} \\ \vec{v} & = \vec{r} \times \vec{\omega} \end{align} $$


*

*To construct a translating motion it is the same as above but with the rotation axis at infinity $\vec{r} \rightarrow \infty$ and the rotation speed at zero $\omega \rightarrow 0$


$$ \begin{align} \vec{\omega} & = \omega \vec{e} = \vec{0} \\ \vec{v} & = \vec{r} \times \vec{\omega} = v \vec{z} \end{align} $$ where $\vec{z}$ is perpendicular to both $\vec{r}$ and $\vec{e}$.
So a translation is the same as a rotation at infinity with a finite tangential velocity.
Related: https://physics.stackexchange.com/a/174209/392
A: They become equivalent for infinitessimal angular displacement and/or distance to the centre of rotation approaching infinity.
A: This might be too basic/obvious, but just stating it in case it is helpful:
A rotation motion corresponds to a force acting on a body towards a direction perpendicular to its velocity. However, in a translation the force is acting parallel to the velocity.
