Many engineers and musicians use the two terms interchangeably. But if you want to get tweaky about it, echo effects are a subset of delay effects, but delay effects can be much more than just echo. A basic description of delay is that it captures the source signal and time-shifts (delays) it by a preset or user-configurable amount of time, then plays it back one or more times. Delay effects range from slapback to ambient washes, and can be used to create unnatural, other worldly sounds through self-oscillation, variable mix and decay controls, complex rhythmic repeats, modulated delays, multiple lines of delay, and so on.
Echo effects are also time-shifted repeats of a signal, but they are designed to sound like the echo found in nature. Many include ambiance to increase the natural effect.