Hitch Size: 2" vs 1.25"

I'm putting a hitch on my car so I can take my bike to some state parks in the Spring. UHaul will install a 2" or 1.25" hitch with or without a "hidden crossbar". I know little about cars and towing.

How badly will I restrict my rack choice by opting for the cheaper 1.25"? I won't be using this for anything but bikes.