nanomsg library is MIT-licensed. What it means is that, unlike with ZeroMQ, you can modify the source code and re-release it under a different license, as a proprietary product, etc. More reasoning about the licensing can be found here.
ZeroMQ API, while modeled on BSD socket API, doesn't match the API fully. nanomsg aims for full POSIX compliance.
The library is implemented in C instead of C++.
In ZeroMQ there was no formal API for plugging in new transports (think WebSockets, DCCP, SCTP) and new protocols (counterparts to REQ/REP, PUB/SUB, etc.) As a consequence there were no new transports added since 2008. No new protocols were implemented either. The formal internal transport API (see transport.h and protocol.h) are meant to mitigate the problem and serve as a base for creating and experimenting with new transports and protocols.
Please, be aware that the two APIs are still new and may experience some tweaking in the future to make them usable in wide variety of scenarios.
One of the big architectural blunders I've done in ZeroMQ is its threading model. Each individual object is managed exclusively by a single thread. That works well for async objects handled by worker threads, however, it becomes a trouble for objects managed by user threads. The thread may be used to do unrelated work for arbitrary time span, e.g. an hour, and during that time the object being managed by it is completely stuck. Some unfortunate consequences are: inability to implement request resending in REQ/REP protocol, PUB/SUB subscriptions not being applied while application is doing other work, and similar. In nanomsg the objects are not tightly bound to particular threads and thus these problems don't exist.
Internal interactions inside the nanomsg library are modeled as a set of state machines. The goal is to avoid the incomprehensible shutdown mechanism as seen in ZeroMQ and thus make the development of the library easier.
One of the long-standing problems in ZeroMQ was that internally it uses BSD socket API even on Windows platform where it is a second class citizen. Using IOCP instead, as appropriate, would require major rewrite of the codebase and thus, in spite of multiple attempts, was never implemented. IOCP is supposed to have better performance characteristics and, even more importantly, it allows to use additional transport mechanisms such as NamedPipes which are not accessible via BSD socket API. For these reasons nanomsg uses IOCP internally on Windows platforms.
One of the aspects of ZeroMQ that proved really confusing for users was the ability to integrate ZeroMQ sockets into an external event loops by using ZMQ_FD file descriptor. The main source of confusion was that the descriptor is edge-triggered, i.e. it signals only when there were no messages before and a new one arrived. nanomsg uses level-triggered file descriptors instead that simply signal when there's a message available irrespective of whether it was available in the past.
nanomsg implements priorities for outbound traffic. You may decide that messages are to be routed to a particular destination in preference, and fall back to an alternative destination only if the primary one is not available.
DNS queries (e.g. converting hostnames to IP addresses) are done in asynchronous manner. In ZeroMQ such queries were done synchronously, which meant that when DNS was unavailable, the whole library, including the sockets that haven't used DNS, just hung.
While ZeroMQ offers a "zero-copy" API, it's not true zero-copy. Rather it's "zero-copy till the message gets to the kernel boundary". From that point on data is copied as with standard TCP. nanomsg, on the other hand, aims at supporting true zero-copy mechanisms such as RDMA (CPU bypass, direct memory-to-memory copying) and shmem (transfer of data between processes on the same box by using shared memory). The API entry points for zero-copy messaging are nn_allocmsg and nn_freemsg functions in combination with NN_MSG option passed to send/recv functions.
In ZeroMQ, simple tries are used to store and match PUB/SUB subscriptions. The subscription mechanism was intended for up to 10,000 subscriptions where simple trie works well. However, there are users who use as much as 150,000,000 subscriptions. In such cases there's a need for a more efficient data structure. Thus, nanomsg uses memory-efficient version of Patricia trie instead of simple trie.
ZeroMQ has a strange double-buffering behaviour. Both the outgoing and incoming data is stored in a message queue and in TCP's tx/rx buffers. What it means, for example, is that if you want to limit the amount of outgoing data, you have to set both ZMQ_SNDBUF and ZMQ_SNDHWM socket options. Given that there's no semantic difference between the two, nanomsg uses only TCP's (or equivalent's) buffers to store the data.
Finally, on philosophical level, nanomsg aims at implementing different "scalability protocols" rather than being a generic networking library. Specifically: