The steepness occurs when the neutralization point is reached. And it occurs early on in the case of weak acid-strong base titration because the acid is weak, and less base is required to neutralize it.
Another point to be noted is that weak acids do not dissociate to large extents as in the case of strong acids. In a solution of a strong acid, almost all of the $\ce{H+}$ ions are already liberated and very few remain connected with the anion (say $\ce{A-}$) as $\ce{HA}$. On the other side, in the case of weak acids, only a few $\ce{H+}$ ions are released and most of them remain as $\ce{HB}$ (let $\ce{B-}$ be the conjugate base of the weak acid)
So lets see what happens as you add the base to these solutions:
Case 1: Strong acid
Once the base is added into the solution, the $\ce{OH-}$ ions combine with $\ce{H+}$ ions to form water. The $\ce{OH-}$ ions continuously deplete the $\ce{H+}$ pool in the solution, increasing $\ce{pH}$.
case 2: Weak acid
Here, it's similar to the previous case, but there's a catch: There still remains some undissociated acid molecules in the solution. Once the $\ce{OH-}$ combine with $\ce{H+}$ and remove some of these ions from the solution, the equilibrium of dissociation of weak acid is shifted to the right. The undissociated acid molecules now dissociate and replenish some of the $\ce{H+}$ ions lost due to base addition. This makes the change in $\ce{pH}$ quite less as compared to the strong acid case.