Database migration program, that can analyze source and target db, and tell diff I'm looking for a program that can help with migration of both database schema and data from a source db, to a target db.
And it has to be able to do it while target db stays live, all the time. So clearing target db, and doing a full migration, is not an option.

We usually use Navicat SQL Server for this, but I'm just wondering, if there is any alternatives out there.

It would also be nice, if the program can make a database diagram, for the SQL servers that does not support the way SSMS does this. (like azure db's)
And unfortunately SSMS isnt up to this task.

What would a working alternative be to using this program?
There are multiples alternatives out there. I would recommend these two:

Idera comparison Toolset and the Red Gate SQL Compare &amp; Red Gata Data Compare. All of them offer full funcionality during the trial period so you can take a look and decide which one is better for you. They work in a similar way: you select source and target databases, compare and generate deployment script with the differences.
For database diagram you can also use the Red Gate Dependency tracker. 

I use Redgate ones everyday and used Idera in the past so let me know if you have any issue.