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 ?