A match cut refers to a cut in a movie where something between two scenes matches something else in the next scene. It is typically used as a transition between two scenes that are closely related. An example of a match cut would be someone lighting a match in one scene and then the sun appearing at the exact location of the match in the next scene. Often, the most important element of a match cut is that the shape of an object matches the shape of a second object after the cut, and both objects are in the same place in the frame.
A cutaway is a type of cut that ends a scene abruptly. Recently, it has come to mean a cut that introduces a scene that takes place outside of the time line of the film and then returns to its regular time line. This is not necessarily the classical definition of a cutaway, which is simply any transition that suddenly takes the viewer to a different point in time.
A match cut and a cutaway are not necessarily mutually exclusive. It is possible to have a cutaway that is also a match cut. Both of the cuts tend to be abrupt and used at the end of a scene rather than in the middle. They may have similar effects on the viewer because they both represent a similar ellipse of time.
A cutaway mostly refers to a cut that advances the story, whereas a match cut refers to the mechanics of a specific cut. A cutaway could use a match cut to help create a less jarring experience for the audience when shifting to an ostensibly unrelated scene.