I enjoy playing both bows, but have been leaning toward German lately. Each bow has its strengths and weaknesses. I started in a public school program, which commonly teach French bow, since most bass students previously learned violin, viola or cello. I didn't study German bow until I was a junior in college, but I really loved it. For me personally, the issues come down to:
- I find German bow more effective at transferring arm weight into the string, but more challenging when it comes to string crossings, especially on the low side of the bass.
- I find French bow far easier for string crossings, but it produces more hand fatigue for me, especially at the thumb joint. (The older I get, the more I notice it.)
You'll find that the effects of both bows are the same, but produced with a different set of muscle mechanics. Whichever bow you play, learning the other will help you become better at your "native" style. If you're serious about playing the bass, I'd strongly encourage you to learn both.
Hope this helps! Here's a well done video that illustrates both bows: