(1) It uses a different elliptic curve than Bitcoin for signing (EdDSA, which uses Schnorr signatures on a Twisted Edwards curve).
(2) It uses a different hashing algorithm than Bitcoin for PoW, which is AES heavy and currently performs similarly on GPUs and CPUs. One of the main downsides to this is that sidechains are currently impossible (validation takes too long), however as sidechains don't actually exist right now we've been ignoring this. If we want to add sidechain support in the future, the hashing algorithm can be change to something simple. In the meantime, the algorithm is relatively "egalitarian" in that no specialized hardware is required.
(3) One time use addresses ("stealth addressing") is mandatory for all transactions. This makes light clients very difficult to secure or create in general, but it dramatically enhances privacy because it's impossible to ever reuse an address.
(4) All transactions are denominated in base 10, and fractionated by mantissa.
(5) Ring signatures obfuscate spending of outputs by allowing you to do a 1-of-N input for a transaction where you spend funds from Bob OR Alice OR Michael OR Claire OR et cetera. Like one time use addresses, this is a passive privacy technology that doesn't require any active participation of anyone in the network (unlike DarkCoin, CoinJoin, and so on).
(6) A single pair of private keys is used for the recovery of all outputs owned by a wallet, but with a different type of data structure than BIP32 has (viewkey/secretkey).
(7) An implicit, silent multisig implementation centered around Schnorr signatures is being researched and developed (thanks andytoshi/gmaxwell).
(8) Research is ongoing into ways to break our privacy technology and improve it. See: https://lab.monero.cc/
(9) Monero is readily auditable from a regulatory perspective (you can easily prove your ownership of funds if you need to, for example to tax agencies).
(10) It has a much faster emissions (subsidy/reward) curve than Bitcoin. 80% is mined within 4 years. The emissions curve is also much smoother than for Bitcoin, with reward decreasing every block.
(11) Unlike Bitcoin, Monero will have long term perpetual inflation. Subsidy will become fixed in about 10 years time at a flat rate of less than 1%, to keep the chain from becoming fully deflationary and to better incentivize miners. This makes it more likely to be useful as a currency than Bitcoin