The question seems fine to me. Off the top of my head:

1) The Jacobian is  a group, and in fact an abelian variety, whereas the curve usually isn't. This gives you a lot of structure to play with that you didn't have initially. For example, to show that a general curve doesn't map onto a curve of smaller positive genus, you can use the fact that the Jacobian of such a curve is simple.

2) The Jacobian is the motive of the curve, loosely speaking. In particular,
all cohomological information about the curve can be read off from its Jacobian.
Eg. Etale cohomology $H^1(X,\mathbb{Z}/n)$ is just the group of $n$-torsion points (up to twist if you're a stickler). I believe that Weil first constructed the Jacobian in the abstract setting precisely for this reason.

3) It has not just one but two universal properties. It's the universal abelian variety the curve maps to, and it's also universal parameter space for divisor classes of degree $0$ (i.e. it's both $Alb$ and $Pic^0$).

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