Everyone else is having a go, so I might as well too;
Pain in your hands, arms, shoulders, neck etc all stem from having too much weight on your hands. The reason you are ok mountain biking is that you sit back a bit more on the MTB.
Your saddle does look too high, which also encourages you to tilt forwards.
Remedy; saddle down and back.
This will also extend the reach on the bike, and you might benifit from a shorter stem until you get used to it.
You are aiming for a position where, when on the hoods or on the drops, you can let go of the bars without falling forwards.
Lose all the padded stuff as well; by making your hands comfy, you are encouraged to take more weight on them.
You should be able to ride with both index fingers as your only support on the bars; not trying to do a handstand..