write <up> a contract

"write" in The Britannica Dictionary:

2.

b : to produce (a written document, agreement, rule, etc.) by writing

• She wrote a contract for me to sign.

— often + up

• She wrote up a contract for me to sign.

What's the difference between "wrote" and "wrote up?"

One word. An unnecessary one.

The addition of the particle 'up' emphasises that the contract was in a finished state.

In certain contexts, there's a difference. Let's say I'm in a lecture and I want to take notes. During the lecture, I'm "writing/making/taking notes". When I get home, if I want to make my brief scribbled notes into something fuller/clearer, and from which I might actually be able to revise the subject later, I'd "write up my notes". In this context, using "write my notes" in my previous sentence wouldn't work. Adding "up" makes it clear that I'm turning my contemporaneous notes into something more complete.
In the same vein, if I needed to write a report for work, and I'd been making a few preparatory notes in the previous couple of days, I would probably have those notes to hand in order to "write up my report" before submitting it.