> What would be the difference between those two things?

The difference between those things is who the police actually do anything for. Who gets (violently) arrested and who gets away with it is strongly connected with your position in social order. Simply put, steal $20 from the register or a loaf of bread from the store and the state will pony up tens of thousands of dollars to try and arrest you, detain you, prosecute you, and punish you. But if you decide to steal overtime from your employees, what do you think the chances are of you finding yourself in cuffs? It's estimated that wage theft involves three fold as much theft as other kinds of robbery combined; why is violence necessary to stop 1/4 of theft, but not the remaining 3/4?

Take a look at protestors too, and see how the police change treatment based on the content of the protest. It's been shown that cops are 3x more likely to react violently to left wing protests than right wing protests in general, and 3.5x more likely to attack peaceful left wing protests than peaceful right wing protests. That is not protecting society, that is protecting the social order.

> As in, what would a police who’s job it was to “protect society” do that current police do not?

Processing all the rape kits they have sitting around would be a fantastic start. Nationally the clearance rates for all kinds of violent crimes are abysmal, maybe cops should stop focusing on grifting overtime and do something about that?

More importantly, a cop whose job is to protect society should stop doing things that are actively harmful to society. Such a list would include but is not limited to:

Not shooting unarmed people, not choking a handcuffed man to death, not tear gassing whole city blocks after being ordered not to by their bosses, not pulling over black people at unusually high rates, not destroying homeless camps, not destroying cooling centers for the homeless during a historic heat wave, not lying to children to make them confess, not beating a family near a protest and then pretending that "antifa" abandoned the child at a "riot", not arresting citizens that they ran into while driving recklessly, not lying in court repeatedly and constantly to get convictions of the poor and powerless, not abusing people in their custody, not raping people in their custody, and not protecting other cops that blatantly break the law.

Cops in the United States have effective blanket immunity for anything short of calmly strangling a handcuffed man on camera[0]. Not only is there no effective means for them to be prosecuted, given the close reliance DAs have on cops, but there is also a specific carve out for them to be protected from civil liability that is absurdly protective of them. Simply put, unless if a cop does something that has specific precedent for, you cannot sue them. And very little precedent can be made because nobody can sue them in order to create precedent. This gets down to the level of "sending the attack dogs against a surrendering person in the woods is a violation of their rights, but not in a 'grassy ditch' because there is no precedent for that in a ditch. Case dismissed". A police force that is supposed to protect society would itself be expected to actually follow the laws that they are nominally enforcing, not be given special protection from illegal conduct.

> Or vice-versa, would such a police still protect from theft or violence?

Do you believe that cops actually protect you from theft and violence? How soon do you believe a cop will be there if someone breaks into your house in the middle of the night? Cops are really there to punish people after the fact, not to protect you[1][2]. How much effort they'll put into actually solving the crime is very much dependent on who you are and what happened to you. Ask a rape survivor how sympathetic and helpful the cops were, if you can stomach such stories.

0 - Make no mistake, Derek Chauvin would still be a cop if that had not been video taped. The official police press release after that was that a man died due to a "medical incident" in their care; a blatant lie. Makes you wonder how many citizens cops have murdered and blamed it on drugs or "medical incidents".

1 - Well, that and arrest random people to find drugs and steal money from citizens to fund their own department.

2 - They in fact have no legal obligation to try and protect you from violence.