I don't have jawbone, I only have the fitbit One and I have to say I love it. I like it so much I gave one to my sister - who is actually using it, I am thrilled to say. AND she said that it is helping to motivate her to move more.
The fitbit one Pros:
keeps track of:
flights of stairs
also shows you how much you've moved recently - by the number of petals on your flower, the more petals, the more you've moved
the website - at least the older version, that I am currently using, breaks your activity level up into
sedentary, lightly active, moderately active, and very active - I really find this helps me to move more, as research shows the less sedentary the better
the clip that comes with the fitbit has this little piece of rubber on the end which seems to come off easily, and when you try to put it back on, doesn't stay on. This leaves these 2 exposed metal pieces, which I have found damage some of my pieces of clothing that I clip it to. I got the fitbit in Dec and have had 2 of these clips lose this little rubber cap.
The "sleeve" that comes with the fitbit to use when sleeping has terrible velcro-like seal. It opens up easily and seems to lose it's "stickyness" quickly.
Doesn't have a good interface with most phones, only recent versions of iphones and galaxy? I don't know which ones because I don't have either. I mostly use the connection to my computer and the readouts that I see on the One- this is sufficient for me, but may not be for everyone.
I also have a bodymedia core (like a bodybugg) that I wear still (the subscription is still active so I continue to wear it). I like the fitbit more in some ways and there is no monthly fee!
Hopefully others will chime in as well.