Depends really, they can be synonyms, or they can have completely different meanings, it all depends on context, as with everything in the English language.
So we have "brighten up" - which can refer to maybe telling someone to be happier or whatever, in that case, it's a synonym for "cheer up",
you could say "brighten up the room" as in literally someone turning on the light and brightening up the room (It doesn't always have to have a special meaning, sometimes it can be quite a literal description),
you could refer to something literally brightening up, like "The wall really brightened up after painting it" as in it was painted a brighter color than before
And then lighten up could also be a synonym for "Cheer up", same as "brighten up", and it's the primary usage of the phrase, like if you ask anyone, that's what they'll say the phrase is used for, so yeah.
Now there is an uncommon expression of "lighten up", where someone loses a lot of weight and someone else tells them "You really lightened up, didn't you?" But as I said, that's very uncommon, still, felt like throwing it out there
Anyway, i think i rambled enough, hope i explained everything, if you'd like to have a class with me or anything of the sort, you know where to find my profile, see ya