Strictly speaking, for just Internet browsing you will NOT notice a difference between wired and wireless. Internet speeds in home are usually between 15-30mbps, while wireless G operates at 54mbps, which means that even on just wireless G your Internet connection isn't even saturating wifi. Where wired connections really show their worth is local LAN file transfers.
Wireless. A router, by definition, makes a Broadband Internet connection wireless. So there is no such thing as a "wired" router (although you can use it wired), because all routers are wireless. A modem isn't wireless, it connects via Ethernet. A router does the opposite.