I've made things out of both materials. Starboard is solid thus much more dense. Use it if the item you are making requires strength or screw holding ability (providing the screws are attached properly). Starboard is not paintable.
Azek is neat stuff. It is a PVC product and you can get it in 4' x 18' lengths. It is white and can be painted with a solvent based paint. It takes a beautiful Imron finish like theres no tomorrow. You can also glue it together with pvc cement and pressure. The faces are smoother than starboard and the core is a dense pvc foam. Its also fillable and repairable with bondo. Its not as strong as Starboard but will hold up well in certain applications. Just as starboard you can also rout the edges (1/4 round, chamfer, etc.)
If you want to seal the edges of azek use pve cement. You may want to mask the face on both sides and edges so you don't get the pvc cement on those surfaces.
Kommacell (also pvc based) is neat stuff too with an almost glass finish and available in widths from 1/8" to 3/4" . its not as easy to get as Azek which is available at most lumber yards.