Pure mathematicians study patterns in ideas which typically arise from other ideas; I'm not a pure mathematician. Statisticians study patterns in data; I'm not a statistician. Applied mathematicians of type 'E' use mathematical tools to study the world around them; I am not an applied mathematician of type 'E'. Applied mathematicians of type 'M' discover interesting mathematics lurking in the real world around them; I call myself an applied mathematician of type 'M'.
A very general theme in what I like to do is the idea of "seeing differently" as well as "seeing further". For example, treating biological questions mathematically still retains a whiff of heresy, as does thinking seriously about the artistic and aesthetic qualities of mathematical knowledge.