The remakes are undeniably easier than their original incarnations.
I'll compare Super Mario Bros. and New Super Mario Bros. to answer the question. NSMB is a lot longer than SMB, and it's much harder to 100% the game (partially because you can't really 100% SMB). Some levels in NSMB are more tediously challenging than any in SMB and can cost you lots of lives. NSMB provides way too many chances at 1-Ups, balancing out those tedious levels. In two levels you can have as many as you'd collect after three worlds in SMB. I've never gotten a game over in NSMB (unless I did intentionally, to see what it looked like... don't remember if I did). I have, of course, gotten better at older Mario games; I can beat SMB without getting a game over, too. But, I could have beaten NSMB with much less experience than needed to beat SMB. Saving makes newer games worlds easier than older ones. Super Mario Advance 4: Super Mario Bros 3 had the saving system right. You should be able to save anytime, but not continue from that save point indefinitely. Obviously, some saving has to be involved for a long game like NSMB to be beaten at all...
Conclusion: If there weren't so many freaking 1-Ups, NSMB could be harder than SMB. I like a good challenge, and hope Super Mario Galaxy delivers... I understand that easy games are more appealing to anyone, and Nintendo aims for the whole family, but what about us hardcore fans? Hard mode, please!
In the case of Super Paper Mario... I died here and there, got stuck at some places because I'm stupid and blind, but the boss battles were cake.
Edit: This topic has made me want to play Super Mario World. I have one goal to find in this save...