Eh, in 25 years of flying, I've never gotten into or heard a strongly worded conversation about mil vs civ pilots. I think that's one of those things that gets exaggerated. Half the time I couldn't tell you what the background of the other guy in the cockpit was.
There are differences, but mil or civ is just way too simplistic. A military pilot comes from a known training pipeline, but there are a million ways to make a civilian pilot, some great, some not so much. The military pilot is subject to a kind of "up or out" training philosophy. A civilian pilot can buy himself a lot of proficiency if he has the cash.
On the other hand, a civ guy who flies for food at a commuter might get close to a thousand hours a year flying seven legs a day into some majorly congested airports. The mil fighter guy will be lucky to get a couple of hundred a year. The mil transport guy might fly more than the fighter guy, but he will probably fly longer legs with much of it spent on autopilot and much more of it spent not as the pilot in command, which is also true of the commuter guy.
On the third hand, the fighter guy will be flying the most high performance of the aircraft, mostly by hand and, after winging, always as the only guy with controls. He will also, by far, have the most experience in dynamically maneuvering an airplane. That's not a big deal unless you wind up upside down or in a nasty stall, which was enough of a factor in enough accidents to warrant the introduction of various "unusual attitudes" training courses to teach pilots who have never been upside down, that the plane still actually flies that way.
On the fourth hand, fighter guys have less experience in multi crew cockpits, though they actually have more experience than most people give them credit for since fighters almost never work alone. As for other facets of Crew Resource Management, mil guys may tend to be more rigid in the application of rules, civilians often have a greater ability to think outside the box. Despite people's kneejerk reaction to laud outside the box thinking, like almost any tool it can be used well or poorly.
Someone who can use differential thrust to fly the jet when the hydraulic lines are cut is using his noodle. Someone who routinely presses below instrument minimums to get to the gate on time is not.
The fact that one airline likes mil guys, or another likes transport guys, or a third likes civilians generally means the square root of jack. Delta, who historically liked Navy guys so much that the Delta uniform is actually the Navy SDBs with different buttons, has shown no substantially safer performance than United, who legend has it, preferred civilians. It has also shown no worse.
Personally, I'd have rather had a good civilian pilot in the seat next to me than a bad mil one any day, and vice versa.
JMO