Presumably 16.0.1 corrects for any deficiencies in 16.0. One of these deficiencies could be to correct an improperly implemented application programming interface. This could potentially cause a problem to show up (bad calling arguments working with bad interface declaration). The update notices should specify the changes made.
Yes, there could be API issues depending on the context of the application per-se and is better to recompile with the newer version to be on the safer side. (Thanks Jim for responding earlier).