Central Forces: Newtonian/Coulomb force vs. Hooke's law We know that a body under the action of a Newtonian/Coulomb potential $1/r$ can describe an elliptic orbit. On the other hand, we also know that a body under the action of two perpendicular Simple Harmonic Motions can also have an elliptic orbit. Hence I was wondering if we can differentiate between a body under the influence of a central potential $1/r$ and a body under the action of two perpendicular SHM’s just by observing the orbits without prior knowledge of the potential they are under. So my question is how can we differentiate between these two potentials?
 A: Your two examples are both central forces. For gravity the potential is:
$$ U_g = -\frac{k}{r} $$
while for the simple harmonic motion the potential is:
$$ U_s = kr^2 $$
Both of these allow circular orbits,and for a circular orbit you cannot tell which is which. However for an elliptical orbit you can because with gravity the origin of the force is at one focus of the ellipse with for SHM the origin of the force is at the centre of the ellipse.
As a side note: these are the only two potentials that have closed orbits. This is Bertrand's theorem. The behaviour is also different with respect to the virial theorem. For the gravitational potential the average values of the kinetic energy $T$ and the potential energy $V$ are linked by:
$$ 2T = -V $$
while for the SHM potential we get:
$$ T = V $$
A: 
I was wondering if we can differentiate between a body under the
  influence of a central potential 1/r and a body under the action of
  two perpendicular SHM's just by observing the orbits without prior
  knowledge of the potential they are under.

As a first note, you have described the two motions in different ways:
the former dynamically, the latter kinematically. In the first case
your description points to the kind of force acting, in the second on
motion being a composition of two SHM. Of course you know the dynamics,
as is shown from your title, where Hooke's law is recalled.
A second point is: what do you exactly mean by "just by observing the
orbits"? If you mean simply discovering that both orbits are ellipses,
obviously there's no answer - they are indistinguishable. 
At the other extreme, I assume you don't think of identifying the
center of force, which would give an easy solution. Yet this could be
done by pure kinematics, computing the (vector) acceleration. 
But there is an intermediate way, which uses Kepler's second law (the
law of areas). In Newton/Coulomb case the speed has a minimum at an
extreme of the major axis and a maximum at the other. In Hooke's case
speed at both extremes of major axis is the same, at the minimum, and
maximum is attained at minor axis' extremes. Thus a simple measurement
of speed would give the answer.
