Mirror-ball approach: Pros: One render, full spherical coverage, directly usable result so no required transformations or stitching (ugh)
Cons: Unless your skybox is for lighting only, you probably want lots of resolution. With only one image for every direction, you need LOTS of resolution.
Also introduces distortion at the edges of the image. This will cause a funky spot somewhere depending on the orientation. The funky spot can be maneuvered somewhere unimportant like straight up or down.
Cube-faces approach:Pros: Spread over 6 faces so getting high resolution is easy for skyboxes.
Because the cube-faces are distributed equally around, this approach vs equilateral, angmap, etc introduces lowest distortion anywhere in the image while not requiring massive oversampling like equilateral images do at the top and bottom. (the entire top and bottom edge of an equi. image represents only ONE point. High quality yes but at a size cost)
-Any program can generate a cube face set because they are just normal renders.
Cons: Usually requires conversion into something usable by other programs, although some viewers and game engines can use cube-face sets directly.
Requires 6 separate images with different camera angles for each, so clunkier rendering process.
This is from experience, please correct me if I'm wrong.
Also, after years of struggling with HDRShop limitations and lack of alternatives, I came across Pano2VR utility. It's not free but not expensive either, and it seems to handle huge images with ease (64-bit version). It will transform just about any panorama to any other, and handles EXRs and HDRs too. In addition to using it for transformations and alternate image-format outputs, it can also generate HTML5, Flash, and QTVR representations for instant viewing!
It's also set up to easily allow as many different outputs as you desire in one 'go' click.
Cons:
-It has a certain bias against .bmp files, but it WILL read them. In Windows you have to enter *.bmp in the selection folder for your source images to appear.
-Undocumented how to get it to input a cube-face set in one go, otherwise you have to manually tie each image to a direction, uck. The format required for auto-loading is to end each cube-face image with a 0 thru 5 digit indicating N,E,S,W,U,D.
-There is also no forum

If it's not in the documentation (which is ok), you are on your own.