What are the basic differences differences between the browsers powered by Gecko and by those powered by Webkit?

Gecko and Webkit are different rendering engines. Yes, they use somewhat different rendering mechanisms. Yes, the results sometimes look different, though that usually indicates a bug in either WebKit or Gecko or both. Also, Gecko-based browsers use SpiderMonkey as their JavaScript engine, while WebKit-based ones use either JavaScriptCore or V8. Also, different browsers use different networking libraries (Gecko-based ones use necko, while WebKit-based ones use a variety of different libraries).

There is a Wikipedia page listing the differences of layout engines: https://en.wikipedia.org/wiki/Comparison_of_layout_engines_(HTML5). One main difference between these mainstream layout engines seems to be that the set of tags they support can be quite different. Though those tags might be seldom used/can be emulated by CSS or JS anyways.