Minecraft – Can the friend and I play minecraft together with different mods

Get and use MagicLauncher instead of the vanilla Minecraft launcher, and use its "profile" manager to set up one profile with mods for single-player and another profile without mods for multiplayer.

Because it loads mods dynamically, you don't have to alter the minecraft.jar, leaving you a perfectly-vanilla game for multiplayer. In your single-player profile's setup just add the mods to the lists of internal and external mods, and the launcher will add them to Minecraft only when you start the game with the single-player profile. When you want to play online, quit and restart to login with the vanilla, multiplayer profile.

Yes, this is doable, but you will need, as ratchet freak pointed out, a separate copy of minecraft for each set of mods. multiMC is one way to do that, but if you upgrade to the new launcher, you can create custom profiles, each with a separate set of mods, to make it even easier (see here for details, since the new launcher adds a couple of steps to the modding process).

Also, once the new launcher is more mature, and Mojang finally get around to providing official mod support, the whole process of switching mods on and off will probably get even easier.