RTMP is a protocol mainly used for live video streaming and interactions. For plain streaming you can use various RTMP servers like Wowza Streaming Engine, Red5, Adobe Flash Media Interactive Server.
You are probably looking for a RTMP host for some script/application that implements live streaming, videochat, webcam recording or some low latency interactions.
Certain apps require specific RTMP side code that is only available for certain server types, you will need to check application requirements. It’s best to ask developers if you’re not sure what hosting type do you need.
RTMP used to be the de facto standard for streaming live video. It’s got close ties to flash, which most browsers have dropped today – few mobile devices ever supported it. It’s using one socket and is streaming at one set quality – it’s implementation dependant what happens when bandwidth is insufficient. Some implementations drop frames, other lag behind.
It still has its use today, as many applications use RTMP for broadcasting. It’s not used so much for edge delivery to viewers, that’s where HTTP streaming comes in.
HTTP, or HTTP live streaming or HLS, uses regular web servers for delivery. A playlist contains URLs to segments of varying quality and the player fetches the playlist every so often and picks segments to play back a continuous stream. This is very easy to implement and scale since webservers and content delivery networks for them are a commodity by now. Pros, it easily scales for the viewer if their bandwidth is insufficient. Cons, latency might be higher as one or two segments generally need to be written to disk before the webserver can serve them.