As already mentioned the difference between the options you mention is really git vs. Mercurial. This cannot be stressed enough.
git is a tool written by exceptional programmers for their personal needs. They want to be able to do exceptionally advanced things, they are confident that they know what they are doing.
If your organization does not only employ exceptional programmers, expect high costs for learning, trouble shooting and frustrated people discussing in the coffee room when using git.
Mercurial gives you 80% of the functionality for 20% of the effort compared to git. (Hmm, such claims might be close to violate Stackexchange policies, I have no real statistical evidence behind these numbers)
For the average programmer the functionality of git and Mercurial are very close to each other. For some special cases git might be superior even from business perspective. If they apply to you, I assume you have git experts in you organization already. If they can convince you, go for it.
I cannot comment about the service quality of the hosters you mention.
Disclaimer: Yes, I do use git and I do like it in certain aspects.