We are running our application on Windows with an MS SQL database, but we are running in Azure IaaS with HA. We had no significant issues migrating our servers into Azure, and performance hasn't suffered*, although there are some issues.
The lag is noticeable, but generally not more than mildly annoying. This may not be true for you depending on the distances involved.
Due to the added complexity of VPN/Firewalls/Hybrid Cloud ( on-premise and Azure infrastructure) we find there are some functions like running reports or performing a change role between HA servers that are best done from a desktop client running in Azure and not on our local laptops. Again, your setup will likely be very different, but you can probably expect some similar issues.
DR is a tough one. Microsoft will gladly set you up with an active/active model where every IaaS server is actively synced between two data centers and you can switch between them. It's not inexpensive. Whatever your current backup strategy is, you will likely find it needs to be modified to work in Azure based on performance and cost. There are some big picture questions to be asked when looking at DR. If a Azure data center goes down, do you invoke a DR response to restore backups in a different data center, only to get 1/2 way through that process and your original data center comes back online?
What are the pro's? I'm told there are budgetary advantages to shifting from capital to operational expenses. Exec's seem to like saying "we are in the cloud". From my desk, it's the exact same thing but in a more complicated infrastructure environment.
* for purposes of some comparison, we run about 13,000 jobs a day in production, with jobs split between on-prem and cloud agents.