It's the difference between what makes Mario a good platformer and LBP a bad platformer. SSB is a game where movement and movement options are possibly the most important thing, precisely because you're expected to be moving all the time. Platformers should be responsive and you should move like you expect you should. Melee gave you tons of options in this respect. Even putting aside 'accidental features' like wavedashing, the game gave you more than enough movement options to get around the stage, and L-cancelling, as much as I disliked the principle of it, minimized the amount of time you were left doing 'nothing' for a second or two because of the landing lag on an aerial (like Link's Down-A). I can't really speak for most of the specific differences in terms of the technical aspects, but Brawl got rid of a lot of the fluidity in the movement - there was no more momentum conservation in jumps (the biggest casualty IMO), characters moved and felt a little floatier, and there was, of course, tripping.
Landing lag itself is basically the amount of time you're left 'stuck' to the ground and unable to do anything after performing a certain move. To take Link's Down-A again, whiffing it means that Link plants his sword in the ground and has to take a bit of time to yank it out and return to 'neutral' stage again where he can do anything else. From a competitive standpoint, it's bad for a lot of reasons, making aerials punishable (so pushing the risk/reward balance more towards risk) and making aerial follow-ups (like Marth's forward-air carry) impossible, because there would be no way to follow up with the 'lag', if it's too long, and your opponent would be able to get out.
From a more casual standpoint, the risk/reward ratio in using that stuff doesn't really change, since missed air attacks (probably like 50% of what you do in an average Smash game) mean a lot of accumulated time 'sticking' to the ground unable to do what you want. It just doesn't 'feel' very good. It makes things feel, I dunno, syrupy.
It's ultimately one of those 'invisible' aspects of the game that no one without any investment will ever particularly notice, sort of like how there's a bunch of CoD players with absolutely no idea what '60 fps' means, but still insist that the game 'feels' better to play than its' contemporaries.
Concerns about landing lag and movement ultimately aren't just purely for the benefit of the competitive community. They're something you can always feel even if you can't precisely define it.