In which ways is OpenStreetMap superior to Google Maps?

One fundamental difference would be that it's hard to impossible to get raw data from the mentioned services. Maybe you could extract it using some kind of script or whatever but then it would still be probably breaking their terms of usage and therefore be illegal to use the data.

To add a second point: With OSM data you have a large legal liberty on how to use the data, especially when not mixing OSM data with other data. For details see OpenStreetMap's licence ODbl or maybe ask at the legal-talk mailing list.

Depending on the country (community) OpenStreetMap can have much more detailed and current map data. The difference on the major road network is only marginal (I feel that rerouted or new highways appear faster in OSM than anywhere else, but I might be biased as it's partially me that updates them ;)), but especially for hikers and cyclists I think the commercial map providers hardly anywhere reach the level of detail that you find in OSM (at least here in Europe - it's probably different for countries that don't have mappers on the ground and low res satellite data).

Google Maps is actually completely unusable here in my region for hiking, because it has a lot of paths that never existed or are not passable in it's database, while other important ones are missing. OSM has (almost?) the full network of local hiking routes as well as a lot of other small paths and landmarks. Same goes for cycling, the Google cycling map improved a bit in the last time but it's far from the level of detail on the OpenCycleMap.

It is great that OpenStreetMap can be used with very few restrictions (see https://www.openstreetmap.org/copyright ) or cost (but note http://wiki.openstreetmap.org/wiki/API_usage_policy if you are using osm.org's tiles).

I am a hiker and I found that it disappointing that i couldn't easily and legally share most maps and routes because of copyright, with openstreetmap i am allowed to share them. I couldn't legally draw a route on a most maps, then pass it to friends as a photocopy, email it or publish it. I know you can maybe do some of this with Google maps, but most footpaths and trails will be missing. If they are missing from OSM we can walk them, record them as a GPX, and add them to OSM. Because the use of OSM data is free lots of navigation apps for smart phones and Garmins have been created that are free or very low cost without OSM they couldn't exist. Well done OSM!

Its mission to show my/your own neighborhood map by me/you not by mercy of others with their interest to earn money on your information.

As your question is very vague, in practicality it lot depends on your application, effort you can put for part of maps work related to your application and last how much control do you want. These three points are in addition to @skorbut point related to pricing/free data.