What are the differences in dedicated hosting vs VPS

Why would I choose one or the other?
how hard is it to host multiple servers on a VPS?

Do you mean Dedicated Server by dedicated hosting or just renting a game server from a GSP?

The difference between a dedicated server and VPS is that with a dedicated server, you get to use 100% of its hardware, while VPS is limited to certain amount of CPU cores, RAM and Disk Space of the machine it runs on.

There is no real difference between a VPS and Dedicated Server except that Dedicated Servers are usually more powerful whilst VPS’s are optimized to suit your needs.
GSP’s who want more power from the bucks decide to go with Dedicated Servers because they are in the long run cheaper than a VPS. Example: 4 VPS’s that can hold 2 servers each (usually) costs more than 1 dedicated server that can hold 8 servers.

As you stated, you want to host multiple servers, yeah? Well, you can either get a Dedicated Server (I prefer Dedicated Servers over VPS) with a good CPU, such as E3-1230v2 (Aim to get a high clock-rate CPU model as SRCDS isn’t multithreaded!). You don’t really need much RAM unless you’re gonna host Minecraft servers. A GMod/Source based server only use about 300-600MB of RAM each.
You can also go with a high-end VPS (Which would be more expensive) and select a few CPU cores (Make sure the VPS’s cores are based on a high-end CPU model aswell) and 2-3GB of RAM. (Burst it up to 4GB if you plan to run Windows Server as it is a more resource hungry operating system than the lightweight linux.)

I feel like I’m getting too much into detail here so I’m just gonna post this tldr.

TL;DR: If you simply want to host a single or two game servers you should go with a VPS. If you plan on hosting a shitload of servers, go with a dedicated server, it’s cheaper in the long run.

In the end, it’s all about preferece.