Confusing pricing, limited hardware choices, and a feature mix that is different.
The king is dead, long live the king!
In our case, a feature mix that is inadequate when it comes to SQL Server. We use the Developer Edition for our Dev boxes... AWS RDS (as just one example) carries the Express, Standard, and Enterprise Editions only. You could stand up an EC2 box for such a thing and management has been advised but management wants only RDS for our environment. I have no idea what the cost differential is but have been told to "not worry about it". We'll see how that pans out.
Also, if you're an xp_CmdShell user (and we use it for a lot of things), forget it. It won't work on an RDS system because they don't make it available. I don't blame them there because they're responsible for the safety of the system and they don't know how it can be used safely like the good DBAs out there know how to.
I also have "scratch" databases that require NO backups (everything in the database is WIP and totally expendable) and so they're set to the SIMPLE recovery model (which also helps a lot with performance thanks to minimal logging for a lot of bulk operations). Forget that on RDS... they necessarily keep all databases in the FULL recovery model because they're also responsible for backups, restores, and data protection in general. Of course, performance will suffer and unnecessary backup space will cost more.
On that same subject, our on-premise systems are setup for (what the NetOps folks call) SAN Replication for purposes of replication to the DR boxes. That means that we don't have to fart around with log files and that means that we can slip into the BULK LOGGED Recovery Model when we do index maintenance to keep log files from being as big as the quarter Terabyte indexes that we rebuild. That won't work on AWS RDS because they necessarily force your databases to stay in the FULL Recovery Model.
It also takes us quite a while to login to AWS. There's a secondary verification where it sends a verification code either to your phone or through email. Our email is in the cloud as of a couple of years ago. What used to be very nearly instantaneous now takes a least a minute and sometimes up to as much as 6. Lovely.
Our on-premise systems are also really powerful... to get the same performance on the cloud requires some pretty high prices levels. I think management is in for a big surprise... we shall see.
I'm sure there's more "fun" limits coming up but, my opinion of "managed systems" in the cloud is that the cloud can be smoke, mirrors, a lot of closed doors, and several dead cows.