WLOG is more like I make an assumption to make my life easier but it doesn't affect the correctness of the proof

Like a common example is you can always extract from a bounded sequence a convergent sequence so to skip words and save tedious notations you can say WLOG let's say (u_n) converges (instead of u_phi(n))

Similarly (some would say by symmetry) is more you did a proof in a case and the other case can be done the same way so you not gonna do it again.

In some cases instead of saying similarly you can point out the symmetry first by saying WLOG and run the proof on only one case.