The Seidel aberration polynomials  was developed in the mid 19th century. But the Zernike polynomials were introduced in the early 20th century.

Here you have a brief comparison:

http://www.sinopt.com/learning1/desnotes/seizern.htm 

Is there are other methods describing aberrations?

Dear Vladimir,

Zernike polynomials represents optimum balanced aberrations in the circular pupil and the expansion coefficients do not show crosslink, as it is orthonormal set of functions (for any other pupil geometries a set of useful polynomials can be obtained). Further, they provide useful optical information of the wavefront. several of these polynomials are well related with optimum balanced third order classical aberrations having minimum wavefront variance, and thus maximizing Strehl ratio.

You can use any other orthonormal polynomial set but their optical usefulness has to be demonstrated. Mathematically, you can expand the wavefront aberration in any of these sets and get an accurate approximation. However, it is true that Zernike polynomials are widely used in the optical community as a standard specification and characterization.

Hope this helps you.

José Antonio