The biggest changes, in my opinion, would be to three things: reach, AoE spells as RangerWickett mentioned, and contrary to what your friend said - slow moving characters.
Most D&D battles - and this is a huge generalization, but I think it's pretty fair to include "most" as a qualfier here - are not played in large, expansive areas, but in small, cramped ones. Castles, dungeons, forest clearings, buildings... These are your battlefields. Ironically, battlefields are not.
This means that fast characters won't care, because they're already fast enough. They can already get all the places they need to go, so it won't really effect them. Slow characters, on the other hand, will have a boost. Suddenly the 20' movement cleric will be able to GET to the person who's down; now the fighter can close on the mage; etc. It won't be huge, and if they're dropping it in 4E the 4E guys obviously didn't think it was big enough to matter, but there you go. Halflings rejoice.
AoE spells will not be square, which is just odd-looking, but that's not really any more odd-looking than the huge cut-out you have to use just to judge them now. In fact, it should speed up AoE placement because everyone will know what it will look like, easily. Big plus. Although AoE spells will also be much bigger, which will give them a boost.
And it will mean that reach will be even more powerful. Now, you won't be able to skate diagonally into a creature with a 10' reach without an AoO. This means, since most players are medium or smaller and most monsters are medium or larger, that monsters will also get a boost. Again, it's not much, but it will be there. (And in 4E they're changing AoOs a lot, so maybe that fixes this balance issue somehow.)
Personally, I don't think I would mind the change, because I don't think it will hurt many things and I think the ease in bookkeeping is worth it. However, I can see it honestly being a bit of an issue in a game where the players are all used to counting a certain way: you're going to end up wasting just as much time early on re-counting as you'll save for quite some time using the new way.