For hiking and biking, I like Windy Maps. It shows terrain contours and highlights foot and bike trail relations, and has an easy-to-use GPS track recorder. You can also ask for future weather forecasts along a navigation route, which is pretty keen if you're traveling long distances. Windy Maps does have a driving navigation mode, but I find that the extra map details that are handy for outdoors activities crowd the map too much for easy reference when driving, especially in dense areas.
For general map browsing, I tend to use Organic Maps. It does a better job rendering urban environments, and even shows primitive 3D buildings when OSM has the data. It's decent as a driving GPS.
Debatably, the best OSM-powered mobile app is OsmAnd, which is a more powerful application with the downsides that come with that -- slow, lots of settings, crowded UI. The killer features IMO are: 1) it's the only one (best I know) that allows users to zoom extremely close, which is vital to see all POIs in dense areas and also helpful to see details of parks and trails 2) it's the only one (best I know) that can interface with Android Auto enabled vehicles. (There's an iOS version of OsmAnd but I've never used it since it requires a subscription for map updates.)
There are particular use cases (eg hiking) when one of these OSM-powered apps will be an adequate or even superior replacement for Google Maps. These use cases are the exception. Google Maps has its own two killer features: 1) realtime traffic for car navigation 2) sophisticated public transit routing, again with realtime data integration.
All 3 of the above apps will attempt public transit routing, but I'd never use them. They don't comprehend train schedules, much less realtime data. They don't handle bus directions at all, though this might be down to the state of the OSM data in the locations I've tried them -- bus route relations are difficult to create and maintain in OSM, and easy to break.
I'm an OSM fanatic myself, so it's saddening not to be able to wholeheartedly endorse any of the current slate of apps. The good news is that the available apps have never been better and continue to improve. Sometime soonish it might be merely a minor inconvenience to de-google your mapping life. (Apple Maps is also pretty good with realtime traffic and public transit routing, if that's an option.)
I tried it in NYC... the map says "Google." If I'm going to be using Google anyway, why not just use the real thing? (This thread lists some possible reasons, obviously.)
I'm fairly sure they don't use Google at all for the transit routing which is their secret sauce. They just overlay their transit routing data over the top of plain Google Maps. Google is pretty horrible with certain routing. For example transferring to following train at a designated transfer station is something Google struggles with. There is a station in my home city designated as a transfer station with timed connections where Google maps will tell you not to transfer to a train 5 minutes later and actually suggests you wait 15/25/35mins for the following train) whereas other transit apps seem to handle this fine (e.g. it can infer that Train 1 is timed to stop at the station 5 minutes before Train 2 at the same station in order for people to transfer and therefore this transfer is suggested to the user as the most optimal route).
I don't know why, but I love Mapy.cz. Someone recommended it to me when I was hiking in eastern europe, and I use it daily for hiking and biking. I don't own a car but I occasionally also use it when I have rented or borrowed one and it works fine.
Here are OSM-based maps applications for Android.[0,1]
Not all OSM-based Android apps listed on OpenStreetMap Wiki, so check out also maps apps in various F-Droid repos.[2]
Also here are maps apps for Symbian[3,4] and Maemo[5].
For other platforms there are also a lot of other apps.[6]
[0] https://wiki.openstreetmap.org/wiki/Android#OpenStreetMap_ap...
[1] https://wiki.openstreetmap.org/wiki/Comparison_of_Android_ap...
[2] https://apt.izzysoft.de/fdroid/
[3] https://wiki.openstreetmap.org/wiki/Symbian
[4] https://wiki.openstreetmap.org/wiki/S60Maps
[5] https://wiki.openstreetmap.org/wiki/Software/Maemo
[6] https://wiki.openstreetmap.org/wiki/Software