Different races/classes have changing speeds - does this affect the distance when using Travel Pace?
No - but some things can change it under specific terms
Travel Pace generalizes the time it takes to go a distance when travelling:
While traveling, a group of adventurers can move at a normal, fast, or slow pace, as shown on the Travel Pace table. The table states how far the party can move in a period of time and whether the pace has any effect. A fast pace makes characters less perceptive, while a slow pace makes it possible to sneak around and to search an area more carefully.
At this point, the faster speeds aren't creature dependent, but dependent on how aware you want to be of your surroundings as you travel.
Mounts and Vehicles provide a faster, but short term option
For short spans of time (up to an hour), many animals move much faster than humanoids. A mounted character can ride at a gallop for about an hour, covering twice the usual distance for a fast pace. If fresh mounts are available every 8 to 10 miles, characters can cover larger distances at this pace, but this is very rare except in densely populated areas.
Characters in wagons, carriages, or other land vehicles choose a pace as normal. Characters in a waterborne vessel are limited to the speed of the vessel, and they don't suffer penalties for a fast pace or gain benefits from a slow pace. Depending on the vessel and the size of the crew, ships might be able to travel for up to 24 hours per day.
It suggests that mounts can help increase your travel speed, but only for about an hour without needing to exchange your mount for a fresh one.
Combined, it seems there is some leeway here, but overall, travel pace is travel pace regardless of base speeds.
This is further substantiated in the DMG on page 242 (Thanks sdjz!)
The difference between walking speeds can be significant during combat, but during an overland journey, the difference vanishes as travelers pause to catch their breath, the faster ones wait for the slower ones, and one traveler's quickness is matched by another traveler's endurance.
If you continue, there are additional boosts to travel pace if a creature is:
is traveling with a flying speed or with a speed granted by magic, an engine, or a natural force (such as wind or a water current)