Both lightning channels and duplex micropayment channels (DMC) are proposals to create bidirectional micropayment channels, that is, micropayment channels that can be used in both directions (a user A can pay to another user B and, at the same time, B can pay to A). Apart from their goal, both proposals also have similitudes in the basic blocks they use to achieve it (e.g. multisignatures, time locks, etc.).
They differ on the exact way to accomplish it and on some of the properties of the created channels. For instance, DMC create channels with a finite lifetime while lightning channels can be kept open indefinitely. They also differ on the exact number of signatures and transactions involved in the creation of the channel, the payments and the resolution of disputes.
A more detailed description and comparison can be found in this recent paper.
AFAIK, both lightning channels and duplex micropayment channels can not be used in practice yet because they need some functionalities that are still not deployed in the main network (e.g., the ability to build upon unsigned transactions). I am not aware of any available implementation of DMC.