Maximum PC

Can I replace the Modem that my ISP provided (RCN) with something I buy?

Yes, but you need to check with your ISP to make sure it's compatible.

And, you'll also need to call RCN to activate the modem.
Cable networks don't use user-authentication. Rather, they use MAC authentication (meaning that they have to enter your modem's MAC address as an authorized device.)

Also, is there any benefit from getting a modem that they didn't give me, or is one modem the same as another?

I doubt you'll be able to do much more with your own modem that you would with the standard issue one. Some modems are capable of higher bandwidth, but your modem isn't your bottleneck here; your account with the ISP is. If you pay for 10mbps, but you get a modem capable of 30mbps, you'll still get what you pay for.

Past that, I really can't think of what tangible benefits could come from purchasing your own modem. Most of the network excitement happens on the LAN side of things.