3-SUM hardness vs lower bounds on the complexity
The reason we need complexity assumptions like the 3SUM conjecture is that we cannot prove meaningful lower bounds in any non-trivial computational models. Our best lower bounds are in bounded depth models or in monotone models, or use techniques such as diagonalization which usually apply only to somewhat contrived problems. Whereas we have exponential lower bounds on circuit size for circuits of bounded depth and for monotone circuits, the best lower bound for SAT is linear.

Since we cannot prove meaningful lower bounds on interesting problems in unrestricted computation models, we use the paradigm of reductions: assume that a specific problem is hard, and conclude that other problems are hard. The most celebrated example is, of course, the theory of NP-hardness. It is easy to criticize this methodology: the hardness assumptions that we make are sometimes controversial. Things are particularly bad in cryptography, where sometimes the hardness assumption made in a paper is a thinly veiled version of the assumption that the construction suggested in the paper works.

3SUM hardness is not a new assumption, but has recently found new interest in the community. It explains the complexity of the best known algorithms for many different problems. As such, it is coming to be viewed as the analog of NP-hardness for problems inside P. As an added bonus, 3SUM hardness makes much more accurate predictions on the complexity of problems: rather than just stating that an NP-hard problem cannot be solved in polynomial time, 3SUM hardness allows us to determine the best exponents (smallest $\alpha$ such that the problem can be solved in time $O(n^\alpha)$) of certain problems.