I've played in groups that did both, the only answer to this question as rpgs are concerned is that it depends on the group, and its usually best established how levelling will occur at the beginning of the campaign rather than changing it mid campaign.
If its about not having curveballs thrown at the gm a better tack is probably talking to the players about how they intend to build their character and what major spell/feat/magic items they want to build to and how they work together, then looking that specific stuff up yourself so you're familiar with it as the gm.
gaining new lvls?
i was then thinking of making the players having to actually have to spend time in game learning new spells and abilities at new lvls. call me old school but i believe that you don't suddenly just get new abilities over night. unless your a cleric, pally or druid then your god just gives it to you. but wizard or sorcerers not just suddenly having new spells at X new level or the PCs suddenly just getting feats. nothing hard just time passes at "home" while they acquire those new powers from books or under tutelage.
The assumption is that you've been doing the background work offscreen in your downtime and the leveling up point is when all that work pays off.
I'm thoroughly in the "Don't use XP, just level the characters up when it's story appropriate" camp, so this question kinda doesn't mean anything to me.
If you think it's a good idea to have them spend some downtime, work it into the plot (maybe your party's being hunted, so saying they need time to rest means they can't just turn around and stomp their persuers after learning a new feat one night).
If you don't think it matters, they can just keep going.
I can see the merits of waiting till the end of the adventure arc to level everyone up - it keeps everyone more interested in the story etc.
Again, if you don't use XP, this is kinda a moot point