ZMQ is much lower-level than AMQP. ZMQ is pretty much a thin layer on top of TCP (or alternatively UDP) which offers basic messaging semantics like Publish/Subscribe. You create ZMQ-sockets on multiple machines and data written to one socket will be forwarded to all machines that have subscribed to this socket. There's several more socket types, for example Request/Response or point-to-point (which is mostly like a traditional TCP connection).

AMQP is much heavier and offers persistence and more advanced routing (and a lot of unncessary cruft). In theory you could build something like RabbitMQ on top of ZMQ (I say 'in theory' because AMQP defines its own wire protocol which is incompatible with ZMQ). AMQP is also server-centric whereas you could use ZMQ in a more decentralized system.