Unless I am really reading the 4e books wrong, I think I know of a number of things that are common in my 3e core books campaigns that are not possible with 4e. Now, this is based just on using the core books from both sets, since it is not fair to compare 4e with the whole gamut of 3e or 3.5e.
One recurring theme in all my campaigns is the EFN, or Evil Foreign Necromancer. This is the powerful evil wizard that creates and controls undead minions to fulfill his nefarious goals. I can't seem to figure out how to create this in the current edition. In 1e, 2e, and 3e it was quite easy.
The tinkering gnome illusionist, or any form of specialist wizard, is not possible. It seems that if you do not wish to be a battle wizard of some form or another then you are not going to have an easy time making your character.
Then there is a Ranger who is not either an archer or a two-weapon fighter. Perhaps one that is a bit of both, or one that relies on the longsword and maybe a shield. If you do not fit into the tightly focused definitions of the current Ranger then you are out of luck.
If you want to make a sorcerer, a non-spellbook dependent wizard, well you just have to wait till they decide to bring out the next Player's Handbook.
Looking at my gaming group, discounting the 3 players using races from Arcana Evolved, I would be hard pressed to recreate the group at all. The Elven Bard is right out. The pugilist Paladin looks like he would be sacrificing a lot of his class abilities if he keeps his vow to defend without causing death. The Cleric of Kord is possible, but she would have to change her world view and her methods of engagement. The noble Elf Fighter who fights with lance, sword, or bow is possible, I think, just not as effective. The Elf who is half Wizard and half Rogue is right out. He uses both classes with equal measure, so the dabbling style of 4e multiclassing would not work. The Elf Monk, the speed demon with Boots of Striding and Springing, is not possible right now.
I know that the goals of some of my players are not possible either. To be a great crafter of magical items isn't an Epic Destiny in 4e. The Paragon Paths and Epic Destinies seem pretty highly focused and limiting, if you ask me. I could be reading it wrong, but that is how it looks.
Oh, I forgot to mention one other character that I do not see as possible in 4e. The Dwarf Potion Master. He's part fighter, part Wizard. He makes all sorts of potions that he carries with him in a bandoleer, pulling them out as needed.
I am sure, with closer examination, I will find other things that will be difficult to handle with the core set that I could do before. I am rather glad I was given the books instead of buying them. Would hate to think I spent money on a game I will most likely not run.