Stofftier
What can the official app do what the website can't
Hard to say since it is proprietary. At minimum it can figure out what device you have and try to gather identifiers unique to your profile. Personally I don't consider this a threat. Most of the data collecting will be what you do in the app anyway.
What kind of nondeterministic problems will occur?
Anecdotal, but I had less issues signing up via the discord app using a VPN, than in the browser without a VPN.
Are there safe 3rd party clients?
There were some, but they've been abandoned. Discord doesn't like alternative clients, either, so I wouldn't bother anyway.