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

1) The Jacobian is  a group, and in fact abelian variety, whereas the curve usually isn't. This gives you a lot of structure to play with that you didn't have initially. Ex. Show that a general curve doesn't map onto a curve of smaller genus. Hint: For such a curve, the Jacobian 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).

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

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