What is the difference between PoW and PoS regarding computations when determining a new block? According to Buterin there are 600 trillion computations done per second using SHA256. He and other sources say that PoS solves the issue of not requiring so many computations but I do not completely understand why. 

From what I understand, PoS also requires the participants or forgers to compute a hash value (which is always the same though), which is lower than a certain target value (which varies with time in seconds, a random factor depending on the current block and the balance). But this also requires some computations, so is it just that due to the more or less fixed time and balance factor PoS requires way less computations than PoW, or is there more to that?
I think you pretty much got it. It's hard to give more than a rough explanation, because there is no standard PoS model. But my understanding is that given the most recent block the only thing that will increase your chance of mining the next block is time passing. If something like timestamps is the only thing to check, the number of possibilities worth any one miner computing is limited.

PoS does not have a difficulty target max, but there is a range of possible answers, say 0-N. The last block is used to compute some new target(not a max, but an actual target), say T. A PoS algorithm will define a stake holder's chances differently, but there will be some function that gives them a range on the spectrum proportional to their stake(M...M+S). As time passes either that range grows, or is recomputed given each new timestamp. When a miner's range includes the target, it is there turn to mine the block, at which point a new target is determined.

00:00 > 0___M...M+S____T_______________N  

00:01 > 0______________T___M...M+S_____N

00:02 > 0____________M.T.M+S___________N &lt; Chance to mine a block

A miner could get better chances by acquiring more stake, but that would require new blocks to be mined for stake to actually transfer. More to your question, a miner could also increase the number of computations they have to do by breaking stake up into multiple accounts, but that would not increase their actual odds. It would just increase the number of accounts they have to check.