Before I do that, though, let me briefly discuss their similarities. They are both essentially two dimensional Minecraft, where it looks much more like a platformer than it does an open world. They are both online, of course, but Starbound is a bit different. There aren't single, randomly generated worlds in Starbound. Instead, there are over four hundred quadrillion planets that have been randomly generated, but everybody is kind of thrown into the same 'universe', technically speaking. To my knowledge this means that the star on coordinates X32, Y71 will always yield the same planets whether you're playing single player or online. In that sense, though, the dismay one may have for always playing in the same universe as everybody else is dispelled due to the fact that you are virtually guaranteed to explore planets nobody else in the game has ever even seen before.
Overall I'd say the core difference between Starbound and Terraria is that in Terraria, I find upgrades to improve myself and my base. I make a home in that world and get used to exploring the ins and outs of that world. In Starbound, I don't get a sense of attachment. You find randomized weapons all over the place (which means that most of them are simply things to sell), and while that system is cool, overloading my inventory with junk that is almost always worse than what I'm currently using isn't satisfying. What's more, the improvements I find are solely for my purpose, and while my starship is my base, it feels less like a home and more like a storage facility. I say this without having found any new crew members and without any ship upgrades at all, however, so take this part with a grain of salt. I'm not making a home in Starbound, I'm just returning to where I keep all of my stuff. It's a subtle but important difference.
Another thing that I would say is an improvement that Starbound has over Terraria is the interface. I must admit the keybindings required getting used to, especially with the left and right hand settings (which is a bit difficult to explain in brevity), but overall I much prefer that system. I like having my 'pickaxe' as a hotkey rather than it being on an action bar, and I like an 'Interact' button. In Terraria, opening doors, chests, and talking to allies is all done with the right mouse button, but since that button is your right hand in Starbound, it's quite jarring to go back and forth. I will say I am currently playing them simultaneously, and Starbound definitely gets my vote in this regard.
Lastly, Starbound has a story. Well, first off, it has quests, which is more than Terraria can say, but in Starbound you know your purpose in the galaxy, and you know exactly what you are trying to accomplish, even if you're not quite sure how to get to the end goal. You have several different races that all have their aesthetic appeal and cultures, which is insane to me because each individual race can have three or four civilization types depending on the type of world you find them on, which is awesome. Because of this, you will always be ready to explore these different worlds, following the main quest as you do. In Terraria, if you don't know the next level of progression, it can be almost impossible to tell what to do next, but you won't really run into that issue in Starbound.
I left out lag and connection issues because my personal grievances with Starbound could simply be related to server issues. That is actually the main reason I'm not playing the game more, because running into that issue constantly is very frustrating. But overall, I'd say that while Starbound is mechanically superior to Terraria, I still prefer the latter. It could also be that I've simply played it more and I'm more comfortable with that game, so it's prone to change, but that is my current opinion. If I were recommending either game, however, I would probably push people towards Starbound more, especially if I was certain that my connection issues were personal and not flaws of game design. Also, I will definitely concede the fact that Starbound looks amazing in comparison to Terraria. Of course, that's due to their age difference, but that fact still stands.
Whatever you enjoy, though, both games are well made. Obviously, they both have flaws, but hey, everything does.