The differences in the cable specifications are meaningful.
Like many people, I had the old POTS cabling (Plain Old Telephone Signal) until I signed up for broadband via my TV cable company. I likely have Cat 5 wire coming into the house. The line for broadband was installed new when we added broadband to the cable subscription, but I expect this is Cat 5. The run goes direct to my cable modem and then I am wireless to the rest of the house including the TV (for Netflix). I have separate analog cable run for normal TV viewing..
As the bandwidth increases, so does electrical noise, not good for data transmission. Not a problem if you lose a few pixels while streaming TV, but if data packets are corrupted, then the packet is meant to be re-transmitted, which effectively means it takes longer to transmit the data, so speed suffers.
The trend for broadband is to higher and higher bandwidths. I started at "only" 1MBps (Mega Bits per second) and now my connection is meant to be "up to" 10 MBps.
If you start to stream TV/movies and another person wants to work on the computer, the extra bandwidth will really help.
I would go with the Cat 6 so that you do not have to worry about a retrofit later.
The cat 6 should have better shielding (for the electrical noise) and to prevent interference to or from other devices in or around your house.