First, for apps that don't use the SIM card (i.e. everything but phone apps and the SMS system), it makes no difference at all. Most apps don't need to know or care how many SIM cards your phone support.
Second, for the remaining apps, it depends on the Android version. Older Android versions didn't have support in the Android framework for multiple SIM cards, so there was no standardised way for a phone app to pick which SIM to use. Each manufacturer did things their own way, and the phone included the manufacturer's own phone app. Only that one phone app (or third-party apps written specifically to work with that phone) would work properly and support both SIMs. Other apps might be stuck with the first SIM and not be able to make or receive calls with the other SIM, or they might not work at all.
As Izzy says, Android 5.1 introduced standard multi-SIM support into the Android framework. Now any phone app targeting Android 5.1 or later can access multiple SIMs on any phone that supports the standard way of accessing them. Of course, it's still possible that the manufacturer continues to use their own system instead of supporting the standard Android one, but most have adopted the new system. You should check before you buy the phone.