When should we use a surface for analysis and when should we use a solid model for analysis?

A surface can't be used for analysis, regardless of what you want to do. As far as i know Solid model are used for analysis. In case of surface, it is thickened first and then simulated.

The aspect ratio comes into play here. You can use a surface if the length is very large compared to thickness (say 10:1).
However for solving the software requires the thickness for obvious reasons like calculating mass (needs volume, hence thickness), etc.
Hence for analysis thickness is essential.
Eg : You can use shell mesh on surface and then give thickness as property in HyperWorks ( OptiStruct, Radioss, Etc,.)

Also accuracy too comes into play when you use 2D. It reduces the accuracy.
In some softwares you can directly obtain Midsurface of 3D objects with some distortion. But these can be cleaned up.

Throughout my career as a stress analyst using finite element methods dating back to the 70's I have primarily used surface elements. I have used NASTRAN (NAsa STRuctuiral ANalysis code}​, Algor, FEMPRO , RISA and other job specific algorithms. This is due in large part to computing power. Stress contour mapping is simplified and results are easier to quantify. Solid Modeling is the current trend with no limit on computational power but I reserve the use of solids for items like axles, spindles, king pins, soil (dirt) heaping and shedding characteristics. The choice of element would therefore depend on what,you are modeling.. I would choose surface elements for all anaysis except those that can only be handled with solids. With modern mesh generators aspect ratio is no problem.