32-bit product keys won't work with 64-bit.
This is not true; the key will work. (maybe you just made a typo) The key is specific to the edition of Windows (Home, Pro, Ultimate). The architecture isn't treated as an edition.
To expand on what Nick and Wally said, all applications in a 32-bit are limited to 4 GB of RAM, including the OS, since a 32-bit integer only holds 2^32 - 1 bytes (roughly 4 billion). Having 16 GB of RAM basically means that you absolutely require 64-bit in order to use your computer to it's full potential.
edit: my post is just redundant
. go with 64