There are iBus apps in both English and Chinese. I wind up using both due to pinyin issues. Neither is particularly good.
All bus apps I've used here are hindered by the need to choose "coming / going" on a route: I suppose you have to choose that somehow, but I have no idea if it's coming or going. Northbound, I can tell you.
One app called Taipei Bus is useful because you can choose a route and it'll show you a drop-pin map, overlaid on what I guess is Google's Taipei map, with little icons showing where buses currently are. This is the one I use most: I'll type the numbers in when I'm at a bus stop, and see where the little red dots could take me.
i84 has an all-Chinese interface. I haven't tried to figure it out. (Well, it's easy enough: you type in a route number. But I'm not sure what to do with that.)
Taipei Metro has MRT only. Best feature is you can look at all the maps that you see in each MRT station; also has schedules of when stations open and close.
Taipei Transit appears to be like Taipei Metro, but significantly more limited: maps of station areas only.
Playing right now with 悠遊公車通 (how on earth did that get on my iPhone?!): actually, this one looks promising. Chinese only, which is manageable for me if I know where I'm going. Map of where you are, with nearby bus stops: click on the bus stop, it gives you a list of the bus routes. Click on the bus route, it gives you a pretty good set of info: route start time, end time, list of stops and how soon until the next one arrives, with a map at the bottom like the ones you see at the bus stops. Seems much more intuitive than the other ones.
I still haven't found an app that easily lets me do want I want: figure out what buses can get me from where I am now to whatever destination I want (MRT station, 101, whatever). iBus technically does this, but last time I tried this it had me taking three buses in sequence rather than walking half a block to take just one bus.