In a barrel roll, the aircraft rotates both in its longitudinal and lateral axes, while in case of aileron roll, the rotation is only about the longitudinal axis.
A barrel roll, image from flightsimbooks.com
If properly executed, there is no change in alttitude in case of an aileron roll, while during barrel roll, the aircraft follows a helical path.
An aileron roll, image from globalsecurity.org
In short, think like this- in a barrel roll, the aircraft goes along the surface of the barrel, while in an aileron roll, the aircraft corkscrews around inside a barrel, with the wingtips grazing the barrel surface.