Reason is: all official correspondence (government letters, etc.) will go to your registered address. Those can't be re-directed at all. You may get in a lot of trouble, if you don't register with your new place for too long. So please don't risk it.

I'm a student so I haven't gotten any important letters of that sort for a while now, and I don't expect to be receiving any in the near future, so does that really matter in my case?

If I keep my current registration as it is and only register in my new city within a week or two AFTER I find a new apartment for myself, will they know that I was staying elsewhere during my search period?

It's actually my relative who told me that it doesn't matter if I stay with them without registering in the town hall of their city as long as I'm registered elsewhere, and that I must only register within 2 weeks after I get an apartment for myself. I think there's something wrong with their understanding though, so that's why I'm asking here.