Why do the subexponential algoriths for the DLP not work for the ECDLP?

Elliptic curve cryptography is much more secure for the same parameters because attacks that work on the DLP do not work on the ECDLP. Why do the attacks fail in the latter case?

"Attacks that work on the DLP do not work on ECDLP" is a rather vague statement, as ECDLP is just a particular case of DLP, on elliptic curves. I suppose that you refer to DLP over $$\mathbb{F}_q$$ for some $$q = p^k$$, $$p$$ being a prime.
The intuitive reason why the DLP is harder to solve over (well-chosen) elliptic curves is that they are our best construction of a generic-like group, id est a group with seemingly no more structure than the group law. It is known that over a generic group of order $$q$$, the best algorithm to solve DLP takes $$O(\sqrt{q})$$ steps.
Index calculus algorithms work by creating a large base of the relation between the discrete logarithm of small prime values. This system of equations is then solved to get the discrete logarithm of each element of the factor base. Let $$(g,h)$$ be the DL instance. Next, the algorithm tries to write $$g^sh$$ as a product of powers of the factor base, for some $$s$$. Once it finds such an $$s$$, finding the discrete log of $$h$$ in base $$g$$ is easy.