Taking price dynamics as a separate subject, the only difference between a binary and standard option is its payout profile. A binary option pays out a fixed amount, while a standard vanilla option pays out a potentially unlimited variable amount. Both options can expire worthless "out of the money." If the underlying instrument moves "in the money", a binary will pay a fixed amount, say $10, while a vanilla option will pay anywhere from $0 to infinity depending on how much the underlying instrument clears the strike price.
