In Chamber of Secrets, we see the Weasleys go with Harry to Gringotts and take out a few sickles and ONE galleon from their vault (which empties it).
Why use a vault for stuff that can be carried around in your moneybag?
In-universe, it seems likely that Arthur's Ministry salary is paid directly into his Gringott's vault, in the same way that most people's salaries (including mine) are paid directly into their bank accounts. Molly is picking up his monthly stipend.
Out-of-universe, it's merely another way of hammering home that the Weasley family is as poor as a church mouse that's just had an enormous tax bill on the very day his wife ran off with another mouse, taking all the cheese.
I don't think that the canon ever specifies directly that the vaults are afforded on the account of present wealth, but rather, wizarding pedigree. It just happens that Weasleys have the latter but (presumably squandered) the former - they are one of the very few remaining pureblood old families.
“I think he would have envied anyone who had a key to a Gringotts vault. I think he’d have seen it as a real symbol of belonging to the Wizarding world. (DH)
“I have visited the Lestranges’ vault only once,” Griphook told them, “on the occasion I was told to place inside it the false sword. It is one of the most ancient chambers. The oldest Wizarding families store their treasures at the deepest level, where the vaults are largest and best protected. . . .” (DH)
As such, the most likely scenario is that the Weasleys have always had the vault (though maybe not always keeping it so empty); and the vaults aren't removed from a Pureblood old family merely on the account that they don't store much in them.
We never came to know exactly how much money was stored in Weasley's vault. As I recall from the second book, it wasn't much, but it was still a pile which might not be possible to carry in handbag (although it can be home).
The answer to your question: Security. A vault in Gringotts was more secure than a handbag or home. Certainly, it'd be an awful experience of losing entire fortune of family while forgetting that handbag in Leaky Couldron (I am not even mentioning robbery or pickpocketing here). The home or any other place also doesn't rule out robbery or theft. So for ordinary people, Gringotts was the best option unless you are Dumbledore who thinks that he can create a more secure protection than Gringotts.