I recently downloaded MultiBit and obviously have my wallet address from that.

I then went to the site Coinbase and created an account.  When I go into account settings and create a new Bitcoin address, I copy my MultiBit wallet address and paste it into the bar and press Create New Address.  Once I do that, the address Coinbase now shows is different than the address I copied.

Is this a problem, or is this just them renaming my address to keep it secure?  If I want to go online to the free Bitcoin sites or have someone send me Bitcoins, do I give them my MultiBit address or the one Coinbase gave me, which is different?

Not sure what's going on here, maybe somebody could explain?  Thanks so much.