Models which use spherical harmonics are called 'spectral models'. The following contains a tutorial:
http://www.ecmwf.int/newsevents/training/rcourse_notes/NUMERICAL_METHODS...

Spectral models have some advantages and disadvantages as listed at the above URL. They are quoted here for convenience:

(a) Advantages

• Space derivatives calculated exactly.
• Non-linear quadratic terns calculated without aliasing (if computed in spectral space or using the quadratic grid).
• For a given accuracy fewer degrees of freedom are required than in a grid-point model.
• Easy to construct semi-implicit schemes since spherical harmonics are eigenfunctions of the Helmholtz operator.
• On the sphere there is no pole problem.
• Phase lag errors of mid-latitude synoptic disturbances are reduced.
• The use of staggered grids is avoided.

 (b) Disadvantages

• The schemes appear complicated, though they are relatively easy to implement.
• The calculation of the non-linear terms takes a long time unless the transform method is used.
• Physical processes cannot be included unless the transform method is used.
• As the horizontal resolution is refined, the number of arithmetic operations increases faster in spectral models than in grid-point models due to the Legendre transforms whose cost increases as N3.
• Spherical harmonics are not suitable for limited-area models.