I know everyone loves "the cloud" these days but it probably isn't the right move unless you need that sort of unlimited scale-out that the cloud can do. Even if you do need that you can probably get away with something like cloudflare sitting in front of your dedicated server.
What you are giving away is:
* disk speed -- what was once a short trip down a wide SATA bus is now a network hop.
* small scale reliability -- amazon cloud servers sometimes just go bad. Can you redeploy your whole stack instantaneously and hopefully automatically?
* cost control -- you know how much that dedicated server is and getting a bigger boat is a pretty fixed cost. Cloud can be cheaper until you get hit hard. Then amazon is happy to keep tuning you up and to keep running up the meter.
If you have a small shared server with a bunch of staticish sites living on them there really isn't much advantage to moving to a proper cloud even today.