Multipath TCP is a way of making what looks like a single TCP connection at the socket layer actually be transmitted over several TCP connections over the network, allowing transparent sharing of bandwidth between links and roaming or failover from one link to the other. It's an actual protocol, designed to be transparent to applications and compatible with all of the hardware and software out there that mucks with TCP connections (stateful NATs and firewalls, proxies, etc) by making each connection act just like a regular TCP connection.
Network coding is a technique for reducing the bandwidth used in broadcast applications or applications over very noisy channels. The basic idea is that rather than sending packets directly, you can send the XOR of various packets (perhaps the XOR of new packets with packets that you already know the recipient has because they sent them, or that the recipient has already acknowledged), and the recipient can do some basic linear algebra to solve for the contents of packets that they haven't yet received based on ones that they have already received. This can allow you to reduce bandwidth usage in broadcast applications, and reduce retransmission round trips over noisy channels.
The problem is, it seems better suited for channels that have those characteristics (noisy and broadcast channels), which describes link-layer wireless protocols, but not transport layer protocols like TCP in general. And to apply it to something like TCP would require fundamental changes in the protocol, changing how ACKs are handled to have them reply not based on a single packet but based on how much information they are able to solve for based on the packets received so far. I'm not sure you could do it in a fully backwards compatible way that would make sense for end-to-end multipath TCP.
And as far as I can tell, the network coding research mentioned is, well, just that, mostly research, while multipath TCP is already a series of RFCs, a few experimental implementations, and at least one deployed implementation.
It seems like this article is just trying to connect this research to the recent story about multipath TCP in iOS 7, when there really isn't any direct connection between the two. It sound more like a fluff piece about a professor's research than anything actually interesting (and frustratingly, it doesn't seem to actually link to the research that it mentions either, nor even provide enough a citation to find it).