Hi Guys,
I am having trouble grasping the differences
between the ADO.NET Entity Framework and
the Dataset designer.

According to microsoft:
The Dataset Designer is a set of visual tools for creating and editing typed datasets and the individual items that make up datasets.
The Dataset Designer provides visual representations of the objects contained in typed datasets. You create and modify TableAdapters, TableAdapter Queries, DataTables, DataColumns, and DataRelations with the Dataset Designer.

Well, does the same thing not happen when
I use the ADO.NET Entity Framework to "mirror" my
database structures?

Microsoft Entity Framework

http://www.asp.net/learn/mvc/tutorial-21-vb.aspx

Dataset designer is an older technology.

The Entity Framework is microsoft's officially recommended data access technology. It is going to be using it in several of its major products moving forward and intends to invest heavily in it.

One thing to add... ADO.NET Entity Framework will allow you to do more than just "mirror" your database structure.  One of the advantages it has is to allow you to map your database to your business objects in something other than in a strict one-to-one fashion

The other two answers are good. I wanted to add that besides Datasets being an older way of doing things, they are resource hogs compared to using entities and generic lists. I switched about 5 years ago and have not looked back since. EF solves a lot of relationship problems or at least makes things much easier, especially when you throw LINQ in the mix. 

Like everyone else mentioned EF is the new way of doings database interaction.  DataSets were great in .Net 2.0, but with the addition of EF and LINQ to SQL, DataSets are IMO obsolete.