of course there are several differencies. The SRTM was initially released as a 90 m-resolution grid, while ASTER as a 30 m-resolution grid. now I think it is possible to download also a 30 m-resolution SRTM. Based on my experience a few years ago, the SRTM has a higher accuracy than ASTER. But I am not sure that this is a general rule. You have to check in your area of interest. You can find examples (i.e. comparisons) in these papers:
Hi, the difference is their spatial resolution. USGS DEMs come in different types, you can also check the vertical and horizontal accuracy to be able to properly select which is suited for your study/research.
On technical side, SRTM 30 meter DEM is supposed to be more accurate than ASTER 30 DEM. The reason behind is the technique adopted for DEM derivation. SRTM uses radar observations to construct DEM while ASTER uses stereo imagery and photogrammetric techniques to extract DEM. SRTM is considered more accurate as radar beam penetrates into the tree canopy to get accurate topographic measurement, while ASTER gets reflection of sun radiations from tree canopy, which later, photogrammetrically processed to derive DEM.
This is a good article with comparison - (Nikolakopoulos, K.,G., Kamaratakis, E.,K., Chrysoulakis, N., - SRTM vs ASTER elevation products. Comparison for two regions in Crete, Greece)
I think I read somewhere that ASTER is better in hilly areas and I don't know if that's true. In my case, ratio between sums of app. 400 values from SRTM and from topgraphic maps (1:25 000) is 0,985. I don't know if errors are changing with elevation, because the highest peak in my case has elevation around 1750 m (app.5719 f).