From a cash perspective, it's pretty much a wash. Technically, the mushroom cave tends to average a little more, but even at an overestimate on the average yield, the profits are less than a single chicken producing mayo daily. Very rapidly, the amount of money you get from it will be entirely negligible. Therefore, we need to consider utility when deciding which to go for.
Mushroom caves can get you some of the rarer mushrooms, including Red and Purple Mushrooms and even Morels and Chanterelle which are normally found exclusively in the Secret Woods (or possibly on the farm if you chose the Forest Farm layout). This means you have inconsistent access to Life Elixirs (200E 90H). Note the word 'inconsistent', it's not super common to get Chanterelle or Morel in the cave, these will likely be your choke point. So don't expect the mushroom cave to be 'I have all the Life Elixirs I want', because that simply won't be true. From a bundle completion perspective, this can net you mushrooms for completing the Exotic Foraging bundle quest easier.
Fruit Bat caves can net you various fruits. The key difference here is that with the Botanist perk, they're guaranteed to be iridium-star quality, which actually pushes their profitability higher than mushroom caves, but still less than what a single chicken yields, even when factoring in the occasional double-yield you get from the Gatherer perk. They can also be processed, although most of them probably won't be worth your time to process. The key here, though, is that you can also rarely get the fruit tree fruits which can be used to complete the Artisan bundle much easier.
For myself? The Artisan Bundle is generally harder to complete than the Exotic Foraging bundle for me, so I go with Bats. Plus, when discussing where I'm laying my paths on my farm with the local handylady for planning out future buildings, I can proudly proclaim: "To the Bat Cave, Robin!".