Simple definitions as far as I've been taught...
Brazing: Joining two or more pieces of material together with a filler material that does NOT fuse with the base metal.
Welding: Joining two or more pieces of material together with a filler material that DOES fuse with the base metal.
That's why people compare brazing to soldering...it doesn't actually melt the base metal like welding does.
As far as the strength of a brazed joint vs. a welded joint, I've been told over and over again by my bicycle and motorcycle frame-building uncle that a properly designed and executed brazed joint is as strong, if not stronger than, a welded joint. "Why are modern frames welded instead of brazed then?" you may ask. From what I've observed watching the old fella brazing tubing together, brazing takes more of a "touch" than welding does and would be much more difficult to automate with robotic equipment. Plus, brazing is just plain hard to do.

On the upside, supposedly brazing will not screw with the heat-treat on CroMo tubing since it's done at a lower temperature than welding.
To quote the Wikipedia page on Brazing: "Brazing is different from welding, where higher temperatures are used, the base material melts, and the filler material (if used at all) has the same composition as the base material. Given two joints with the same geometry,
brazed joints are generally not as strong as welded joints although a properly designed and executed brazed joint can be stronger than the parent metal." There's that "skill" thing again.

I'd love to have brazed joints on the parts I build, but that's a set of skills I sure don't have.

I let my buddy and his MIG welder fuse metal for me and it seems to work just fine.