A Virtual Private Server is essentially a "slice" of a much larger server that is made available exclusively to you. When you order a 512MB RAM Virtual Private Server that ram is yours and yours alone to use and nobody else will ever get access to it as an example. You can think of this as the large server being a pie and your VPS being a slice of that pie. You have full and total access (root access) over your piece of the pie.
Semi-Dedicated hosting is a powerful shared server where there are a limited number of clients placed on the server. You do not have full and total access (root access) to the server and you do not have dedicated resources. The resources on the server are available to everybody on the server on an as-needed basis just as they would be on a regular shared however there are a very limited number of clients which usually results in much higher speeds and resource allowances for when you need it. The added benefit of there being a limited number of clients is that the server stabilizes very quickly - we fill the server to the client limit and then the load and usage statistics plateau giving you a very stable environment for your web site.
When it comes down to it if you need to install something that needs root access to the server you will want to go with a VPS and if you don't need root access to the server but you want a more reliable and stable hosting solution Semi-Dedicated hosting would be the way to go. If you are unsure which solution is right for you don't hesitate to contact our sales department at firstname.lastname@example.org and we will be more than happy to help you make your decision.