This sounds like a daft question, but I'm serious.
Translation and rotation are clearly different -- the symmetry between them is broken by Newton's Laws. But in the Lagrangian/Hamiltonian frameworks, they look extremely similar! The Lagrangians for free rotation and free translation are exactly the same, up to the replacement of some letters. Working entirely with the Lagrangian framework, it's unclear when and where the symmetry breaking happens.
Despite this, there are many clear asymmetries between translation and rotation:
- There is absolute rotation, but not absolute translation. (At least, I believe this is the orthodox position.)
- In space, starting with zero linear and angular momentum, it's possible to change your angular position but not your translational position (you can turn yourself around, but can't move your center of mass).
- In quantum mechanics, free particles can have continuous values of linear momentum but have quantized angular momentum.
I know why the third point holds: localization causes quantization, and the set of possible angular positions is compact, while the set of possible positions is not. In fact, I feel like this is the only difference, a priori, between translation and rotation. In layman's terms, if you keep rotating, you'll get back to where you started, but if you keep translating, you won't.
Is it possible to use this reasoning to extract the first and second bullet points above? If not, what exactly is the difference between translation and rotation?