What is the difference between TE and TM mode?

Hint: First let us understand the TE mode and TM mode. Basically these are the modes of propagation of electromagnetic waves. From waveguide theory it is possible to calculate the number of ways in which an electromagnetic wave can propagate within the waveguide. Transverse modes occur because of boundary conditions imposed on the wave by the waveguide.

Complete answer:
A waveguide is a structure that guides waves, such as electromagnetic waves, with minimal loss of energy. For each waveguide mode there is a cut-off frequency. Below this frequency no signals can propagate along the waveguide.

Transverse electric (TE) modes: No electric field is present in the direction of propagation. These are also called H modes because there is only a magnetic field along the direction of propagation (Here H is used for magnetic field).

Transverse magnetic (TM) modes: No magnetic field is there in the direction of propagation. These are called E modes because there is only an electric field along the direction of propagation.

Note:Hollow metallic waveguides filled with a homogeneous, isotropic material support TE and TM modes. In rectangular waveguides, rectangular mode numbers are designated as \[T{E_{mn}}\] or\[T{M_{mn}}\], where $m$ is the number of half-wave patterns across the width of the waveguide and n is the number of half-wave patterns across the height of the waveguide. Only a limited number of different $m, n$ modes can be propagated along a waveguide depending upon the waveguide dimensions and format.