Currently, Bitcoin addresses and their checksums are constructed from the public key by a using repeated hashing with SHA256 and RIPEMD160. Now I understand the reasoning behind using hashing for constructing the checksum, but why wasn't just the original public key with a checksum added used? Is it just because of the shorter addresses or are there other privacy/security implications of using hashes of public keys instead of just public keys?
It's just to get shorter addresses. Regular public keys are 65 bytes long, which is much too long to be convenient. Compressed public keys are 33 bytes and could potentially be used instead of hashes, though these are a little longer than 20-byte hashes. It also seems likely that Satoshi didn't know about compressed public keys or wasn't comfortable with using them when he designed Bitcoin.
Hashes seem to help against certain attacks (some attacks against ECDSA, for example), though they also open up the possibility of other attacks (such as attacks against RIPEMD-160). It's not clear to me that they do improve security overall.