I have ADSL2 Modem which I got from my service provider, now I want to setup wireless(Wi-Fi) in my home. I went to shop, where I came to know that there are routers which come up with modem also but they are priced lil higher than the only wi-fi routers. Now it is obvious that I should go for only wi-fi one, as I already have modem. My question is, is there any difference between ADSL+router and only router?

I think if I use ADSL+router, the speed will increase lilbit, as modem does modulate and demodulate signals, and router also generates wireless signal, that is time to take conversions is doubled, and if I use ADSL+modem, it will directly convert the signals to wireless, and time will be saved, so the overall speed will increase lilbit. This is what my concept is(Which might be wrong).

What do you guys would suggest me? should I take my current modem away and buy an ADSL+router or I should keep my modem and buy only wi-fi one. Please tell me the difference and suggest me which one I should go with, and also suggest me which company router I should go for.