As far as your web application (in this case, DXA, though that's not necessarily relevant) is concerned it's just a different place to connect the CD API up to so, yes, I think it would just be a cd_storage change.
In terms of relative performance this is harder to answer, as it depends on where your existing database and web applications are running. If you're running on-premise with your db server close to your app server then moving your database to Azure will probably make things a little slower - not because of the product but simply because of network latency. If you're already running in Azure, then those factors will likely be different. I think the biggest driver for customers moving to cloud databases is not speed improvement, but more around flexibility, less maintenance and reduced cost. Of the customers I know that have migrated to cloud databases (mainly AWS rather than Azure) none have had issues with performance.