Sometimes the differences can be quite theoretical. Pure math deals with abstract structures essentially, proving theorems for the sake of mathematics and aesthetics with no applications necessary. Applied math relates to applicable things, modeling real world processes with differential equations, finding ways to solve such equations, finding optimal solutions to problems, coming up with efficient algorithms for various number crunching things, and proving theorems related to all these.
An acquaintance of mine, who in another life was a probabilist by training, got a job in a business school. She did consulting for some finance researcher, on some pricing problem. She came up with an analytical solution to the problem in 2 days, which the finance guy gave to his computer whiz to implement. The program worked, but it took 24 hours to spit out numbers meant to decide whether to buy/sell on the stock market, where things change by the minute. That was her first adventure in the wonderful world of applications. A pure mathematician is happy to have an exact solution, or often just knowing that a solution exists, number crunching be damned. In applications, an approximation that is good enough and involves the simplest computations are preferred, because at the end of the day, someone else has something useful to do with those numbers.