Along with destorying the bad bacteria, antibiotics destroy the beneficial bacteria in our bodies that keep yeast production in check. It's a delicate balancing act and antibiotics throw off that balance. Diarrhea exacerbates the problem because the acidity of it will often cause microscopic tears in the skin that allow yeast to flourish. Diarrhea will also cause an established rash to burn very painfully.
Some are more sensitive to yeast than other. I personally and my youngest son are both VERY sensitive to antibiotics and prolific yeast production. Yeast infections are also made worse by increased moisture (such as a wet or loose stool diaper or a woman sitting around in a wet bathing suit etc.) so that's why his rash is getting a little better when you can keep him drier.
What you need to do is take him to your doctor. They can take a swab of his rash and run a 5 minute test that will tell them for sure that it's yeast, then they will give you an antifungal cream to apply three times a day or so. You will also be instructed to change him the moment he wets or poops (or the moment you notice that he has) and to let him run around with a bare bum as much as possible. If you go to the doctor and I'm right, your son's rash will be much better within 48 hours of starting the cream but make sure you continue it until there isn't a hint of the rash left or it will come back.
Best of luck to you.
Just wanted to add that you should NOT stop the antibiotics without your doctor's permission. Stopping too soon will only ensure that you haven't eradicated the infection from your son's body and will actually make it worse. This is because the bacteria causing the infection will recover and will have had a chance to "learn" the antibiotic and adapt to it. If this happens, you'll need an even more powerful one the next time to halt this infection. This can lead you down the road of antibody resistant illnesses/bacteria which are very unpleasant and very harmful. Always, always, always finish a course of antibiotics unless there is an allergic reaction to it.