I used to say that nerds don't have friends. Geeks have friends, but their friends are all also geeks.
This is no longer strictly accurate. The accepted definitions of the terms have changed over time. For reference, when I was at school, I wasn't referred to as a nerd or a geek, I was referred to as a square. And it had the same insulting, bullying intent.
Today, however, the terms have changed both in intent and meaning.
A "geek" is someone who is unashamed about enjoying pop culture, to varying extents. Some geeks collect miniature figurines, some play lots of games, some cosplay as their favourite characters. Comics, D&D, movies - especially superhero movies - are all parts of geek culture. The characteristic of being a geek is being very interested in and excited by a particular hobby or pastime. Non-geeks sometimes characterise this as "obsession", but the characteristic feature is the subject matter, not the level to which one dedicates time to it. One can be a comic geek, or a boardgame geek, or a superhero geek, or a movie geek. Usually a person is a geek about one thing, or a limited range of things. One does not generally say that someone is a football geek or a basketball geek, even though the level of interest in those subjects can equal or exceed that of any other kind of geek. But the word is slowly coming to incorporate sports as well. Geekery is generally social - geeks go to conventions and participate in group activities like gaming.
I am a geek - I am a tabletop gaming geek, a LARP geek and a sword and medieval weapon geek. I am not a superhero geek or a movie geek, or an anime geek.
A nerd, on the other hand, is someone with a broad general interest in nerdy subjects - maths, science, engineering, philosophy, statistics - academic subjects in general. Nerds are not so much into pop culture as geeks are, though there is certainly crossover. Nerdery is more often a solitary pursuit and nerds are often more introverted than geeks are. Nerds will usually have a broader range of interests, as opposed to geeks, who will frequently specialise.
It is important to remember that there is usually significant overlap between these two categories. One can be both a nerd and a geek. I am a nerd - I am interested in magic and mathematics, and I have an interest in science and philosophy, and I am also extremely introverted. I am also a nerd about some of my geek hobbies - for example, I can rattle off D&D rules without looking them up in the books, and I can tell you the differences between a longsword and a shortsword, with references to historical texts and extant examples of both.
Also, you will find that there is often considerable disagreement about the definitions of these two categories. Arguing about the difference between nerds and geeks is definitely a nerdy pursuit.
To sum up: If you want someone to tell you what the airspeed of an unladen swallow is, ask a nerd. If you want to know what movie that is a quote from, ask a geek.
