Since both nerd and geek are essentially stolen words, I use the following definitions to both define them and to make them useful in actual conversation/argument.
Geeks and nerds have the same types, depths, and levels of interest. They are people with deep, often (but not always) esoteric interests that go well beyond those of the average person. A geek/nerd hobbyist is distinct from a mere hobbyist only in their depth and dedication.
So, what then is the difference between a geek and a nerd?
A geek is someone who has the deep, sometimes esoteric interests, yet shares a common bond with others who have similar depths. An anime geek will get along just fine with a videogame geek or a comic geek. A sports geek can find common ground with a film geek.
A nerd has the depth, but not the common bond. A comic nerd isn't comfortable hanging out with non-comic nerds. A sports nerd can talk shop with a sports fan, but would have difficulty with a manga fan/nerd/geek.
Nerds and geeks bond with other nerds and geeks within the spheres of their common interests, be it anime or MMORPGs or what have you. Anime geeks and anime nerds are indistinct from one another within the confines of anime fandom.
The geeks, however, also bond with other geeks of other spheres. They break bread not just over their mutual love of, say, Buffy the Vampire Slayer, but also over the understanding and commonality of having that deep of an interest at all. The Buffy geek feels a bond with the chess geek, and they can relate to one another.
Nerds bond over an interest. So do geeks. Geeks also bond over having an interest.