Meh. Most of you guys are flat out wrong. I get that you guys are trying to help the OP, but if you're going to be confident in your answers, at least do some research.
This is straight wrong. 100% straight wrong. Most games it doesn't matter at all, but try perfect-attacking DDR with half a frame or a full frame of lag. One frame of lag makes a big difference in games that require precise timing. High level/competitive Mario may be one of those games. It's easy to adjust to one frame of lag if you're used to it, but going from years of zero lag to one frame definitely feels off for most people.
EDIT: Maybe Fantastic-attacking on ITG would be harder to do. Perfect attacking has a fairly big window. Oh and also I mean transitioning between lagless and laggy displays. Obviously if someone only practices on laggy setups, they'll be able to do well in rhythm games with that same lag.
Two things that are affecting you:
1) You are used to a CRT. The screen WILL feel different because it does have more lag. 1-2 frames more. That's assuming you've played a while on your CRT and you're used to precise movements/timings.
2) The screen position makes a big difference. Using DDR as another example, try playing on an arcade cabinet that has a screen right next to your face vs one that's a few feet away from the screen. You can't react the same in both situations.
Many monitors/TVs do but you have to specifically look for them.
Here's a good list:
http://forums.shoryuken.com/discussion/145141/sub-1-frame-hdtv-monitor-input-lag-database/p1
From what I read, the gamepad has at most 2 frames of lag, so anything less than 33ms should have less lag.
The game still plays at the same speed, but your timing is thrown off. Imagine if the game had ten minutes of input lag and every time you hit the jump button, you had to wait ten minutes for Mario to jump. Would that still play the same?
Most first party controllers use pretty much lagless wireless connections.
Play on a CRT. It'll be less laggy than the gamepad.
Wavebird might have lag; I'm not too familiar with that. I know that newer wireless controllers are pretty much the same thing as wired controllers though.