The following assumes you live in the US.
It all boils down to what your Television source is, free Over-The-Air (OTA) broadcasts, Basic (unencrypted) cable (no set top box required), a higher cable subscription that requires a set top box, or satellite.
For OTA broadcasts, you need a TV Tuner that accepts a digital (ATSC) signal.
For Basic Cable, you need a tuner that accepts a ClearQAM signal.
For a higher Cable subscription, you could still use a tuner that has a ClearQAM signal, but the signal would have to come from the set top box and I believe the Coaxial output of the set top box is converted from digital to analog. A cablecard solution would probably be better suited.
I don't believe there are TV Tuners for satellite signals.
The only real difference I see between the two tuners you've listed is that the AVERMEDIA H837 AVerTV Volar Hybrid Q HDTV & FM Radio - Black
has the additional FM Radio and ClearQAM tuners as well as some additional functions.
Let us know what your TV source is and we can then make recommendations.