Can my friend and I play minecraft together with different mods?

It depends on the mods- Some mods are client-side only, and can be used without others needing them. An example of this would be minimap mods such as Zan's or Rei's, UI mods such as NEI/TMI, or shader mods. These mods don't add any new blocks or anything that your friend will be able to see- They simply change things on your client.

Generally, if a mod adds new blocks or entities (such as tools or monsters) then it will only work if both you and your friend have it installed. Other than that, a lot of mods will still work without your friend needing to install them too.

No, you can only play multiplayer if you have the same version of Minecraft and the same version of mods.

It is however fairly simple to install something like Feed the Beast and have multiple copies of Minecraft installed at the same time, this way your friend can have mods to play multiplayer and when he's playing alone he can play vanilla Minecraft without having problems with his save files.

As @dlras2 mentioned, it can work if you only have mods that are client side, like mini maps. Basically anything that changes crafting or adds new blocks/items will not work without having the other player also having that mod (because all that is sent is "Hey, make this item" and the unmodded client will then go "I HAVE NO IDEA WHAT TO DO AND I AM PANICKING" and then probably crash)

It depends on whether they are client or server side mods. Something like NEI ( Not enough items ) or a minimap is a client side mod. These need to be installed only on the client. They are things that add UI (User Interface) Elements.

Anything that adds items or mobs is a server side mod. These need to be installed on both the server and the client.

So the short answer is maybe. If you do not know which they are, please ask. Hope this helps.