|My question is which connection should I use for faster broadband speed?|
Regardless of whether you use the wired or the wireless interface, it can't improve your broadband bandwidth. Here's why:
Wireless N = 100 Mbps (LAN)
Wireless G = 54 Mbps (LAN)
Wired Nic = either 100 Mbps or 1000 Mbps (LAN)
Broadband Internet = maximum 10 Mbps (Internet access or WAN)
(typically it's 1.5 to 6 Mbps but 10 is availabe in some areas to home users)
Since your internet connection is a fraction of your LAN connection, the only way to increase internet bandwidth is to get a higher bandwidth package with your provider.
As to which is better on the LAN, the answer is the wired NIC. Wireless is inherently susceptible to lag and latency issues you don't get with a wired NIC.
So even if you had Wireless N and were connected at 100 Mbps, a wired 100 Mbps connection will outperform the wirless every time.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.