Both are similar, and the very short version is they are sandboxed applications that bundle their own dependencies and can update out of band with your distro’s software repository. With Flatpaks they can share a common runtime environment, but I think with Snaps they bundle everything into the snap (I might be wrong about this).
One key difference is that Snap is basically only on Ubuntu, and Snaps can also bundle CLI applications or server software. Flatpaks are currently really meant for desktop applications.
In both cases you can modify the permissions of the the programs they bundle sort of like how you might expect on iOS or Android. That is to say you can restrict their access to the file system, the network, or other things. So, as an example, I can run a proprietary program as a Flatpak but ensure it cannot access my Bluetooth if for some reason I feel that need.