Every MintChip has an ID, and every transaction is logged on both the sending and receiving device with the ID of the other device. This means that if someone takes your chip, they get a complete record of every transaction you've ever made. In other words, it's not anonymous at all. That's problem two. Bitcoin solves this by encouraging users to generate a new address/private key for every incoming transaction, so that matching up addresses to people is hard.
It's tied to single physical devices which can be lost or damaged. This makes them unsuitable for storing savings. Bitcoin wallets, on the other hand, can be backed up securely.
Both MintChip and Bitcoin can be stolen if the attached device is compromised. Bitcoin is designed in a way that makes it possible to fix that, and developers are working on a fix: multi-signature transactions (so you have several computers, or a computer and a phone, and all of them must agree to any outgoing transaction). MintChip, however, cannot solve this problem in any way except with chargebacks, and the documentation given so far indicates that they aren't supporting that.