Why didn't Princess Jasmine use her wealth to help the poor?

That's the basic rich liberal mentality, never use their own money to support a cause just steal from the middle class to do it all.

I think Jasmine was naive and didn't think to bring money since she usually gets everything by command. Also, maybe she doesn't get to handle money since she could have anything she wanted within the palace. I see it more of a naive mistake on her part (she maybe could've asked her handmaid to give her some coins).

But towards the end of the film she does become Sultan and has expressed desire to help others. I don't even think the original animated movie addressed that before.

Probably because even though she's surrounded by luxury, she's got no "wealth" of her own. In heirarchical or patriarchal cultures all the family money belongs to the husband or father, or head of the family. Family members who are young, dependent, and/or female, may have money spent on them, but have no control over how the family money is spent, or the state treasury. Even Jasmine's jewels may legally be the property of her father or the state.

Also because she's led such a sheltered life, shut up in the palace, that she has no idea what kind of poverty exists outside her door. If that was the first time she'd snuck out of the palace and seen what kind of lives the poor lead, then hopefully she'd do better from then on.

The idea (better reflected in the cartoon) was that Jasmine has never been outside the palace walls and had no idea you had to pay for things. The message that sometimes gets lost is Jasmine starts out as a spoiled unknowing royal and becomes a more caring understanding person throughout the story. This is better expressed in the 2019 movie as there is more time to delve into it.