Most vaccines are a protein (originally a fragment from the (inactivated) virus, now more often a synthetically manufactured fragment of the virus).
These vaccines (Moderna & Pfizer) are delivered as the RNA that encodes a fragment of the virus (that your own body then uses to produce a fragment of the virus). The delivery mechanism is very different, but the viral fragment that your body sees is the same as it would if the protein was delivered directly, as is traditional.
RNA can't reliably survive on a surface for more than a few minutes (which is somewhat good news since coronovirus itself is an RNA virus), or more than a few days unrefrigerated, and is very sensitive to any contamination by bacteria or other organisms.
Traditional proteins are generally much less fragile both chemically and as food for other organisms. And sometimes a protein (vaccine) can be specifically engineered to be even more stable in room-temperature, dry, or other unforgiving conditions.
And none of this takes into account specific formulations, or other chemicals that are used to aide the delivery process. Much of this has to be measured empirically. It's not clear to me if there is some chemical difference between the Pfizer & Moderna vaccines, or if the different temps are just what they use as their protocol filed with the FDA.
Further, see /u/dnautics below, often RNA therapeutics aren't completely true RNA (they can have chemical features to enhance stability).
And no, stability testing happens pretty early on. Pfizer would have definitely collected enough data at this point to say whether or not their vaccine is stable at higher temperatures.
Pfizer is big enough to have/let a PR department do a press release during business hours. Of course given that Pfizer did a press release first Moderna for PR reasons needed to release more data to make a bigger splash.
No matter how you look at it, there is important data that will be in the FDA (and equivalent in other countries) submission that we don't have. There could be a "thats funny" thing burred, though odds are against it.
Pfizer hit their interim analysis point before Moderna did, but only by a few days.
The difference is not very big, but to those who like numbers it is big enough.