Well Vietnam wasn't a conventional war between warring states. It was a civil war with the US supporting the South side. It's considered a police action, as no congressional (or governing body) support is needed and doesn't meet the criteria for war. As it doesn't follow the doctrine of total war but was limited military involvement. (see gulf of Tonkin resolution) It is dictated by the ruler of a country or other circumstances and police actions are usually partaken if the action is expected to end quickly, issues with sovereignty of the combatants (there is no actual "recognized" body to fight usually a rogue nation or an organization below the nation state level), or for humanitarian reasons as a defensive action (to prevent a smaller conflict from spreading getting out of hand or spilling over to other areas), or ya know if the president just wants to (/s). The idea is a police action is something that needs to be done right now to prevent undue loss of life and can't wait to be bogged down in the red tape of congress.

The involvement in a coalition of nations (like the UN) can also dictate if a conflict is a war or police action. Treaties require you to commit troops even though your country does not want to, Its a police action.

Most modern wars the US has been involved in, the US has not been the main belligerent but is only a support of the main combatants even though the US may have the largest force in the theatre. This is the same for Russia and China.

It's a way for major powers to prop up or fight in a conflict while avoiding official conflict with another super power (see proxy war) as most have entangled treaties that would draw others into the conflict (see ww1 and WW2)

Note: this is just my understanding of the 2 terms, the use is a bit of a hazy area, probably intentionally, to side step international laws and keep the security council from killing each other.