Rex himself gives the reason

Under this directive, any and all Jedi leadership must be executed for treason against the Republic. Any soldier that does not comply with the order will also be executed for treason.

Ahsoka wasn't a Jedi anymore, but she had been aiding the Jedi in the Clone Wars (before and after leaving the Order). More importantly she had just been talking with the Jedi Council. Yoda also contacts her and she implies she might be willing to rejoin the Jedi (but not yet). So if they are all now regarded as traitors, that makes Ahsoka complicit in their treachery.

Why did Rex want to kill Ahsoka after receiving Order 66?

Order 66 was to kill the Jedi.

Ahsoka was still a Jedi. She may have no longer been a member of the Jedi temple/order, but she still had all the same abilities as a Jedi.

It is no different than calling Luke Skywalker a Jedi. He was a Jedi even though in his time the Jedi/temple order were long gone.

A Sith Deals In Absolutes

Order 66 was created by Darth Sidious, who is, notably, a Sith, so when we try to understand what form it took, it helps to look at how Sith think.

Darth Vader summarizes it best:

If you're not with me, then you're my enemy.

Sidious is not interested in the subtleties of Temple politics. He is not interested in their sects, their offshoots. He doesn't care if you're an infant in the creches of the Temple or so old that you totter around the Archives. If you are a Force Sensitive that has ever been associated with the Temple, you are a Jedi, and you are a target.

The only way to not be a target is to actively align yourself with the Empire and the Sith. Neutrality is not acceptable. By a Sith definition, Ahsoka was a Jedi. Having a tiff with the Council and leaving the Temple didn't change that.