This would-be quick advice became a long answer. Sorry.
As tyler pointed out in his nice answer, calling Dispose() is a great programming practice. This is because this method is supposed to "rally together" all the resource-freeing needed so there are no unneeded open resources. If you wrote some text to a file, for example, and failed to close the file (free the resource), it will remain open and no one else will be able to write to it until the GC comes around and does what you should have done.
Now, in some cases there will be "finalizing" methods more specific to the class you're dealing with, like StreamWriter.Close(), which overrides TextWriter.Close(). Indeed they are usually more suited to the situation: a StreamWriter's Close(), for example, flushes the stream and the underlying encoder before Dispose()ing of the object! Cool!
However, browsing MSDN you'll find that even Microsoft is sometimes confused by the multitude of closers and disposers. In this webpage, for instance, in some examples Close() is called before the implicit Dispose() (see using statement if you don't understand why it's implicit), and in one in particular they don't bother to. Why would that be? I too was perplexed.
The reason I figured (and, I stress, this is original research and I surely might lose reputation if I'm wrong) is that Close() might fail, yielding an exception whilst leaving resources open, while Dispose() would surely free them. Which is why a Dispose() should always safeguard a Close() call (sorry for the pun).
MyResource r = new MyResource();
try {
r.Write(new Whatever());
r.Close()
finally {
r.Dispose();
}
And yes, I guess Microsoft slipped on that one example. Perhaps that timestamp would never get flushed to the file.
I'm fixing my old code tomorrow.
Edit: sorry Brannon, I can't comment on your answer, but are you sure it's a good idea to call Close() on a finally block? I guess an exception from that might ruin the rest of the block, which likely would contain important cleanup code.
Reply to Brannon's: great, just don't forget to call Close() when it is really needed (e.g. when dealing with streams - don't know much about SQL connections in .NET).