What are some similarities and differences between electromagnetic waves and mechanical waves

The similarity is they all follow a partial differential wave equation, usually second order. The differences mainly arise from the physical reasons for the wave.

Mechanical waves arise through energy being imparted to the medium, and when one part is displaced, it pushes or pulls on an adjacent part, and the energy density moves with the phase velocity (the velocity at which constant phase moves through the medium.)

Electromagnetic waves involve a burst of oscillating electric field, and as it oscillates, it generates, following Maxwell, an oscillating magnetic field, which in turn generates further electric field oscillations.

I hope this helped. I have really had to use this for anything since I learned it but I hope it helps you.