Unless you are quite close to the poles, you will be able to ignore that we are talking about spherical coordinates.

The angle in north/south direction can be converted from degrees to meters.
The angle in east/west requires you to also compensate for how close you are to a pole. At the equator, one degree represents a similar amount of meters as the north/south angle. At the north, or south pole, any measured angle will end up representing zero meters (if talking about the geoid pole and not the magnetic pole).

When you have your data as delta-x and delta-y distances, you can perform standard math operations for closest distance between a line segment P0->P1 and a point. Or closest distance between two line segments - one from measured points, and one from expected path.

If you do need to operate close to the poles, you can start by perform a coordinate transformation first, to switch into a different coordinate system.

The thing here is that when trying to measure the distance to a route, you are talking about very small distances. So at the point of interest, you can ignore the curvature of earth, and see your little circle of interest as flat. Of course, you may also decide to use the GPS in 3D mode, and perform three-dimensional matching. But the complexity will increase and this may affect the required processing power. Also, the vertical precision can be quite low unless you are using a more expensive GPS that is using correction data to improve the measurements.