Vector tracing is the way to go for recreating simple artwork. Art that has a lot of gradients and details (IE screen printed pinball backglasses) cannot be vector traced, unless of course you are insane. Think of the Pole Position cabinet art... that is a perfect candidate for vector tracing. Race Drivin' looks a little tricky, as there are some gradients and same details in the tire, but definitely doable.
Both Photoshop and Illustrator have vector tools. You can export any vector tracings you do in photoshop to illustrator and vice versa. It should be in an illustrator format (.ai or .eps) for printing.
In both applications, you use the pen tool to make the actual vector tracing. It is a point, click, and drag process that allows you to create complex shapes. The vector shapes you create are always editable, so it's not like using the paint brush that just throws pixels down on the canvas.
There is no advantage or disadvantage to using either software for making the vector shapes themselves. I prefer using photoshop for making the vector shapes, as the pen tool seems more accurate. However, illustrator should be used when applying the color to the shape, as it will store the applied color as part of the shape (Photoshop can do this too, but it's more refined in illustrator). I've been dying for them to just combine the damn things.
Once the vector is completed, it can be scaled to any size without any loss of quality (as they are computer generated algorithms that you basically create, as opposed to pixels, which get stretched and noticeable at the edges). Then it is a matter of making sure the vector is scaled to the right size so it can be printed on an adhesive.
Hope this helps, let me know if you have any more questions!
Brendan