Yes it is, but it does not tire
The phantom steed can not fly, as it
uses the statistics for a riding horse, except it has a speed of 100 feet and can travel 10 miles in an hour
A riding horse has no fly speed, and so the phantom steed has no fly speed. Since it does not have a fly speed, and the spell does not state the steed is unaffected by difficult terrain, it is affected by difficult terrain on the ground.
Why then does it have an increased overland travel speed?
Several examples in the DMG with increased overland travel speeds can not fly. We can observe that a steam powered contraption does not fly, nor does a sailing boat. But they still do not travel at the normal rate, because they do not tire. The same applies to the phantom steed. The quoted section mixes two different reasons for increased overland travel speed:
- Flying, as air does not have the types of obstructions found on land
- Not tiring, as the transport is powered by magic, wind or an engine
In the case of the steam contraption the engine does not tire, in the case of the sailing boat, the wind does not tire, and in the case of the phantom steed, the magic does not tire. This is stated explictly on page 243, further down in the same section:
Similarly, a phantom steed spell creates a magical mount with a speed of 100 feet that doesn't tire like a real horse.
There is one issue with the rule: if either not tiring or being able to fly can result in increased overland travel pace, should the combination of both (as provided by a flying carpet) not have a pace that is even larger? The rule does not say so. Maybe that was a design decision to keep the rule from getting too complicated. However, you could use the carpet to fly for 24 hours instead of the normal eight, which could result in a tripled overland travel pace (thanks to Ryan C. Thompson for pointing this out).
PS. Issues with the special travel pace rules
For what it's worth, the travel rules in the DMG are a bit wonky to begin with. While they maintain that this is travel at a special, faster rate, the formula to calculate that rate (p. 242/43 DMG) is:
- In 1 hour, you can move a number of miles equal to
your speed divided by 10.
- For daily travel, multiply your hourly rate of travel by
the number of hours traveled (typically 8 hours).
A normal medium-sized character who can tire and can not fly has a speed of 30. According to the Travel Pace table (PHB p. 182) that character's normal travel pace per hour is 3 miles and their daily travel pace is 24 miles -- exactly the same as calculated using these rules for special pace.
So that calculation does not result in travel pace that is faster relative to movement speed than what anyone could get, while it is presented as if it would be due to special advantages. The rule does lead to faster absolute travel pace if you have higher movement speed, as by the normal rules, travel pace would be independent of movement speed. There is however no mechanical speed up for the ratio of normal speed to travel pace at all.
The PHB itself also states that normal roads are no difficult terrain and would not slow you down:
The travel speeds given in the Travel Pace table
assume relatively simple terrain: roads, open plains,
or clear dungeon corridors. But adventurers often face
dense forests, deep swamps, rubble-filled ruins, steep
mountains, and ice-covered ground—all considered
While that is at least consistent with the achieved travel speed, it means that the reason flying creatures are faster cannot be difficult terrain in the technical sense. It must be that there is no direct route, that the overland route is winding to avoid terrain features, while the distance as the bird flies straight is much shorter. Such may not be the case when you use a well constructed road such as the one connecting Neverwinter and Waterdeep along the Sword Coast.
What is more, these speeds are too low compared to real-world benchmarks. Using the rules, an eagle (fly speed 60) could travel 6 mi/h or 48 mi in a normal day of travel. However, in the real world an eagle can fly on average 100 miles in a day, or up to 250 miles when migrating. So the speed should be at least twice of what is given by the rule.
If we infer that the travel pace should be doubled because there is no difficult terrain to overcome for fliers (that normally would half the speed), or because there is no exhaustion effect from repeated dashing for creatures that cannot tire (as suggested in the chase rules on page 252 DMG), then to be internally consistent, the formula for creatures not tiring or being able to fly should be movement speed divided by 5 in miles per hour -- double the pace given. And possibly the rate for a flyer that also cannot tire should be movement speed divided by 2.5 in miles per hour - four times the pace given. All of that would be house ruling. (We use double pace in our home game, and had no issues with it.)