Ahhh, I see. You've got a raster image and you want to turn it into a vector image, is that right? I thought you were just starting from scratch.
Ok, if you just trace around a logo in Photoshop, then yes, only the actual path that you drew will be exported to Illustrator. You could just turn that path into a clipping path in photoshop (it's in the flyout menu on the paths palette) and save your file as an EPS. Then place that file in Illustrator and you'll have your logo neatly cut out. However, the content of the logo will still be a raster image, except it will have a nice sharp outline.
If you want to convert the whole logo to vectors, then there are a number of ways to do it. You could use a program like Streamline to do most of the work for you, but the results won't be perfect. You'll probably have to open it in Illustrator and tweak it a bit.
Alternatively, place the original Photoshop file in a new Illustrator document (without the Link option so you'll have a more detailed image to work with). Now create your paths like you did in Photoshop, only try drawing all the paths, not just the outline of the entire logo. You can then color each part as you go along, although you might need to hide certain areas to see what you're doing. A bit time consuming, but the results tend to be good.
Try out the various options, and see what you're happy with. If you've a lot of logos to get through, maybe the clipping path method is the way to go. If you do use this method, however, there's no real advantage to bringing it into Illustrator unless you're doing a full page layout too - the Photoshop EPS will work just fine.