In most situations: No, it won't. If you're connected by LAN and simply plug in both interfaces, even if you manage to tell the NAS to send data over both it will not grant you a speed boost. The simple question is: what is the limiting element? As mentioned in comment, often the HDD is it. Sometimes the transfer rate of the connection (not really on a lan, but maybe if your connection goes through WLAN) limits the transfer rate. But if it is already used to its maximum capacity, how should it go faster if you have two connections? In most cases it will even get slower.
So what is the different port good for? - A second port is great. If you have a more complex network. From the fact that your NAS has two ports i assume it's not a cheap one, with a lot of functions. For example you can build up two LAN networks, one with your private computers at home where you share your picures and music and one where you like to experiment or even host some serious web services. It's nice to seperate those two networks. Your NAS - being very powerful - now can provide functions you'd like in both networks, for example data sharing or vpn acces. You can now choose to which network which functions to apply. E.g. hosting your favourite music on the private site and some linux kernel source code on the other one. Now you got some friends who enjoy coding and you like to grant them a vpn acces to your - let's call it "dev" - network. You do so with your fabolous NAS and they can become a full member of your dev network. For your family, you set up a vpn for your private network where they can always view their favorite photos.
So to conclude, it's mainly for security / infrastructure reasons. Hope this explanation was not too long.. =)
1are you sure you are limited by Network and not by your HDDs when doing data transfer operations? Two ports are useful, you might want to connect it to multiple physical LANs. – phoops – 2014-04-19T10:49:27.180