Do I need to buy Mac OS X Lion before purchasing Mac OS X Mountain Lion?

No, you can go from 10.6.8

http://www.apple.com/osx/how-to-upgrade/

The end result will be the same.

I imagine pretty much the same. It is possible some feature may be present in Lion that Mountain Lion will abandon, but I can't be specific. If know, for example that I still have Quicktime Player from OS 8 because it still works (in Classic) for movie editing and after OS8 Apple decided to charge for its editing features (playback was free). Some people have also mentioned retaining Snow Leopard versions of some applications because Lion changed the menus (and apparently is changing parts of it back again). I don't know if that's enough to merit buying Lion separately, but it's an observation.