So, if everyone is essentially at full HP and such all the time with only 15 mins of rest, you essentially have to make all encounters harder. If they always have a haste, fly, fireball, righteious might, healing, etc. then I no longer get to wear them down with encounters that are fun for them to hack through and must make more of them potentially deadly.
There are a number of underlying assumptions in this statement which may or may not be true:
1. All players enjoy hacking through encounters that are there just to wear them down.
2. All of a character's resources are regained after a short rest.
The first is a matter of taste. The second is a matter of game design, and while I suppose it is possible to design a game in which all of a character's resources are regained after a short rest, that is completely different from allowing a character to regain some resources after a short rest.
Also, it messes with my timeline if I want to impart a sense of urgency and get the PCs to push on at their own peril. It basically makes it so they can always push on regardless of resources, fights are always full powered and makes one shot or charged magic items nearly useless. Also, duration of spells hardly matters because in 15 minutes, you're all back up to full and can cast everything again.
There are again some more assumptions here that may or may not be true (apart from the "recover all" vs. "recover some" assumption that has already been pointed out):
1. There are no short deadlines.
2. All fights are against foes of equal difficulty.
For the former, deadlines measured in days are obviously less relevant when resources can be wholly or partially regained in a matter of minutes. All this means is that when deadlines are used, they have to be shorter. No doubt, this places some restrictions on the type of scenarios that can be run (as the adventurers might comically put it, "Why am I always told to rescue the princess one hour before she is to be sacrificed? It would be nice if they could inform me three days in advance for a change."). Another alternative, which verges into house rule territory, is to change the length of a "short" rest. Even though the rulebook might define it as five minutes, a DM might choose to extend it to an hour or even a day if t suits his playstyle better.
For the latter, if some fights are against tougher opponents, charged and one-shot magic items will help to even the odds. In addition, on the assumption that an element of luck or chance will still be involved, they can act as safety nets that help the PCs recover from a run of bad luck.