In the old days, most (all?) video game controllers were digital in nature. If you pressed a button, the signal was "on" and if you didn't the signal was "off." Take a look at the Nintendo game controller, and you will see the directional pad in the shape of a plus sign. When you push one nub the signal goes something like "right is on" and the game system deals with it as needed. On or off, up or down, that's the way the signal used to be.
I think it was the original PlayStation (or maybe the Nintendo 64) that introduced the analog stick to the game console world. Now, the controlling signal sent to the game was something like "up about 20% and to the right about 68 degrees" which makes fine tuning possible in a game. This makes skills based games possible (and yes, keyboard and mouse are probably the best skills based game play control scheme available, but that's not the point). The fundamental difference in the control scheme has a massive impact in terms of being able to control the game.
I own a Gameboy Advance (GBA) which is essentially a Super Nintendo in a tiny little handheld form factor. The controls are all digital with a four way directional controller. That controller is actually 8 way if you push the controller to make two contact points trigger. So, top and left points both down means, "diagonally up to the left." Analog sticks offer 360 degrees of control AND a proportion of how much in that direction.
Super Monkey Ball 2 (SMB2) is a skills based game where you guide a monkey in a ball to a goal on the play map. Obstacles, patterned movements of the playing area and randomness get in the way, but using your trusty analog control stick, you can leverage your skills. Without the analog stick, this game suffers immensely. I own Super Monkey Ball 2 for Gamecube (which has an analog stick) and Super Monkey Ball Jr. for Gameboy Advance (which has digital controls). The Gamecube version is far more playable, complex and FUN than the GBA version. It has EVERYTHING to do with the control scheme.
Advance Wars is as good on the GBA as SMB2 is on the Gamecube, and it has everything to do with the control scheme and using to the benefit of the game. Advance Wars is a turn based strategy game that would be suffer it were a Gamecube game, and the producers of Super Monkey Ball Jr should have taken the controls more into account when making the game. The version of SMB for my Sprint PCS Vision phone (a Samsung n400) is better aligned with the control scheme (the dial pad, which is, incidentally, the worst game controller ever) but costs $4 for limited gameplay and actually EXPIRES after a month or so. That's lame.