Hard to say - of the two, I would strongly prefer the GT 1030, as it would perform way better than the 710. However, being a new card, your old HP might not work with it, as Nvidia's newer cards seem to have a tendency to only work with UEFI systems, and not play nice with older Legacy BIOS systems (this is secondhand information for me, though, not through personal experience)
I have an HP slimline as well, and, while it does need low profile, the location allows for the cooler to be a little taller. My Dell 3647 Inspiron Small Desktop, though, puts the PCIe slot all the way at the edge, so the cooler/fan MUST fit within single-slot height. Because of this, I couldn't use a 750 or 750Ti in the Dell. There are regular profile and low profile 750/750Ti cards, but they all have a cooler that extends past single slot in total height, even of the end plate itself is only single slot.
Are you limited to single slot height for the cooling solution as well in your system? I did a quick search, and it SEEMS like a low profile card with a slightly taller cooler would work, but not being 100% sure on my search results, you'd have to look directly in your case.
Any which way, from what I understand, performance from maximum to minimum is 750Ti > 1030 > 730 > 710.
Here's an example of a 750Ti low profile . . note that the fans and shroud give it some extra height. I think it would fit in your system, but would not fit in my Dell.
http://www.overclock.net/products/msi-geforce-gtx-750-ti-low-profile/reviews/7291
However, it seems the 750Ti prices are a little ridiculous given that they're older cards. You may be able to find a good used deal somewhere.
If you can find a GT 730 with the GDDR5 memory, that would be the way to go in your budget. I want to recommend the 1030 instead of the 730, but I just don't know for certain if it'll work in an older system like yours. Older Dells are notoriously fussy about it, but I don't know about HP.