So can someone enlighten me on the differences between the two versions? I only ever played a few encounters of either system so I didn't really get the feel for either.
Huge.
4e is very popular with DMs because it has pretty transparent balance. In 3e you could have high-level characters with really low AC or low damage or one or more low saving throws... Worse, there was very little help in determining what good values were. Contrast with 4e. It's harder to have bad stats, and even if you do at least you know how bad they are. (Typical 4e PC AC is 15 + level, for instance, 1 point higher than a monster, although obviously this will depend on your role.)
Because monster stats are balanced too (as of MM3 onward), you don't have a situation where a 1st-level PC can have AC so high they can't be hit, or get dropped in one hit, nor do you have PCs who can't hit enemy AC. A 1st-level monster would attack at +6. A PC with very bad AC, say a sorcerer, might have AC 12, and that's only because the sorcerer is pretty badly-designed. A 1st-level wizard probably has AC 14 or 15, since they can use Intelligence to AC, and that's the lowest AC a PC using a well-designed class can have. You could play a 1st-level paladin using a heavy shield and +1 armor might have an AC of 21, which means they'll still get hit 1/4 of the time, and that's playing the literally toughest class out there.
Not that players necessarily need to spend a whole lot of time making the numbers work. Put an 18 in your key stat, take an expertise feat, get your items "on schedule" (+1 every 5 levels) and the math takes care of itself. Spend your character building time becoming
cool instead. Don't spend time trying to make a cool blade bard that just keeps "whiffing".
I like roles. Roles are controversial. Some hate them. One reason I like roles is to keep competition down. In 3e a rogue, fighter and fireball-tossing wizard are all trying to one-up each other when it comes to damage. In 3e, the monk has no clear role, and generally ends up being fast and unhittable but unable to do anything except maybe boringly grapple somebody. In 4e, the rogue is a striker and deals a lot of damage. The fighter is tougher, draws hits, and gets extra attacks if someone tries to ignore his toughness. Put a fighter and rogue side-by-side and carnage flows. If someone decides to hit the rogue, hoping to kill them before taking too much damage, the fighter gets an extra attack. Attack the fighter, and you're attacking a rock who has lots of hit points, AC... and is flanking you with the rogue. The wizard cripples the opposition, but won't end a fight by herself. The monk deals a lot of damage too, but spreads the damage around and can toss people through zones that your friendly wizard set up. It's not going to directly compete with the rogue, if you happen to have one in your party too.
Roles are controversial because, while they make you good at what you're good at, they also make you bad at what you're not good at. You can't do a "switch hitter" fighter/mage in 4e, who uses magic till he runs out, then starts slicing up things very well with his sword. Well, you
could, but you'll only be good at one of these things, or maybe even
none of them. The bladesinger class was an attempt to make this trope work in 4e, but it failed. Mind you, a fighter/mage in a previous edition can't make both work for them at the same time. Pathfinder got it to work with the magus, ironically by clarifying it's role. The Pathfinder magus is a
striker, and due to action economy tricks can cast and stab in the same round. Unlike a 4e magic striker, like the hexblade, it
also has a fair number of tricks from the wizard class. Being 3e-based, it has unpredictable AC and so forth, but if you don't care about the numbers the magus is better than any 4e-equivalent class at what it does.
3e had broken magic. Lots of spells didn't mesh well with the system (eg most of those that either gave huge bonuses to skills or required a very high skill score to escape). Because wizards had such weak defenses, they were given overpowered defensive spells to compensate. It really sucked when a PC got crippled because an NPC mage hit them with Tasha's Hideous Laughter and, due to save DC/saving throw problems, the PC failed and failed to stop laughing every turn. And then some.
Yes, 4e is slow. I like to maintain a realistic view of the edition. Monster hit points inflate faster than PC damage (PC damage is less balanced than most other numbers, so it's hard to measure the problem), plus a lot of the bloat makes PCs tougher than they should be.