Never mind, the moon finally decided to show itself.
This is the best I got with the Tamron, handheld with SR, after about 20 shots, no PP, crop :
300mm, 1/90, F8.0, ISO 200.
I tried many shots at ISO 100, but everything was always underexposed, unlike with the DA 50-200. I also had to use longer exposure, 1/90 vs 1/80 with the DA. The color of the moon is also different. It comes out yellowish, which is not what it looked like to my eyes. I think the color of the DA 50-200 is more color accurate. To be fair to the Tamron, there is much more 3D detail than with the DA, but that's probably because it's at 300mm, and also the moon wasn't full.
I didn't take shots on the same day, but this is what I got with the DA last week at 200mm, 1/180, F9.5, ISO 100, also handheld, no PP crop :
I have to get some sleep now.