Vine charcoal and Willow charcoal
You won't notice a lot of differences between using vine and willow charcoal.
Vine (of the grape) tends to be slightly darker in tone, is physically a little harder than willow, and I believe it has a somewhat finer structure. Vine is thinner and usually a little straighter than willow. The latter can come in a larger variety of thicknesses, which can be convenient when wanting to make broad strokes. Willow also tends to have more knots, which can have adverse effects while drawing (e.g. sudden changes in hardness and breaking in unpredictable ways).
Note that these characteristics can differ even between brands (different production processes, different climes, different wood species, different storage, &amp;c.).
I suppose that historically vine charcoal was used more often in southern Europe (such as Italy during the renaissance), whereas in the north people had more access to willow.

If you can get your hands on it, it might be worthwhile looking into 'Nitram' charcoal.
I have no experience with it - I actually just found out about it - but it is supposed to have deeper colours while being as easy to erase as regular charcoal, and it comes in different hardnesses (H, HB, and B). It is easier to sharpen to a finer point for detailed work than regular charcoal types. It also leaves less dust, which might be especially appealing to people with allergies or sensitivities.