When not used to mean "modern," contemporary is usually used as a noun, meaning one that was (or is) alive or active at the same time as another: "Shakespeare's contemporary, Christopher Marlowe, is not as well remembered today." Contemporaneous is never used in this way; it is always an adjective.
As you note, contemporary as an adjective can mean both "modern" and "from the same time period [as another]," whereas contemporaneous only takes the second definition. Contemporaneous works about the 16th century were written and/or published in or near the 16th century, whereas contemporary works about the 16th century were probably published recently (although not necessarily; there is potential for ambiguity here).