I really doubt it has anything to do with them being vets. In Iraq and Afghanistan you can't shoot until you are fired upon but in the US it's shoot first and ask questions later.
I wonder why the police are given so much more weapon's autonomy than the military?
The police, on the other hand, can easily give in to the immediate fear of someone else shooting first: when they are not careful about not escalating, the worst thing that is likely to happen to their side is having to call in reinforcements and having some more bad press that won't add much to whatever they already have. If you want to put it in very crude terms, they are playing war in easy mode and there, many things that are very important to the military are completely irrelevant.
The US military is far more centralised and (despite its many faults) aware of planning for wider strategic issues.
Many police depts don't seem to want to face up to the fact that the tactical problems they have now have root causes in their own short sighted policies, incentives and behaviour. If they carry on like they are, these problems will only get worse.
Other western countries with more centralised police hierarchies seem to put more effort into community relations and other more strategic stuff, even if it means less 'effective' tactics ie less control, arrests and convictions etc.
I think because there is no political will to protect the usual victims of police brutality—indeed, the "tough on crime", and concommitant "tough on people who are in the general vicinity of people perceived to be criminals", mentality, is perenially politically popular.
Of course there is also no political will to protect the usual victims of military brutality, but they can and do (in many cases, to some extent, at least historically) protect themselves, and so it has been perceived that it's necessary to reckon with the possibility of reprisal.