So I see the DirectX and OpenGL specs listed on GPU pages all the time. e.g. this one says DirextX 11 and OpenGL 3.2 http://www.newegg.com/Product/Product.aspx?Item=N82E168...
My question is, do these two specs make a big difference? And can they be upgraded? For example, if a GPU is listed with OpenGL 3.2, can it be upgraded to OpenGL 4.1?
The "DirectX" version and "OpenGL" version is directly tied to the hardware, so you could install the software that "supports" later versions, but it will have to run in "backwards" mode. In a lil' more technical words: the software will use the proper render paths your video card can use, but all the new shinny or whistles won't be available.
actually many older cards already support features of newer openGL and it only takes a driver update to enable them. This was because openGL was slow to release the new version but some cards were build ready for it as the same features were in dx11. Although, yes, they have to have hardware support for the features.