You've got good timing; I'm retired military and now work for a contractor. My daughter's an aero engineer who worked for NAVAIR as an intern and now for a contractor (and we've extensively discussed opportunies). I've also compared salaries for entry level, and typical mid level positions.
Here's my 5-cents, staring with: I consider three company types, not two:
(1) Manufacturer
(2) Private support services contractor
(3) Government Agency
Salary: entry level nowadays is significantly lower for the gov't, but the retirement and benefits are typically better (especially "how many days/year you are paid to stay home" - vacation, sick, gov't shut down, etc). However, advancement and salary increases in the gov't will be rapid (if you're good), and I'd say salaries are probably equal with about 5 years experience. With greater than 5 years experience, the edge goes to the gov't. BTW, my main "competitor" who I lose employees to is the gov't. This hasn't always been the case, but it is today.
Type of Work: This is extremely dependent on specific examples, but IN GENERAL: A manufacturer will provide the opportunity to do hard-core detailed engineering, often have good training programs, but you'll be one of a very large team, and can easily get pidgeon holed. Private service contractors can only offer you opportunity based on current contracts and taskings, and typically very little to no funding for training. The right gov't agency (such as NAVAIR for aviation, or NAVSEA for ships) will not typically provide the opportunity for significant hands-on detailed engineering, but will provide great opportunity for getting a wide range of experience not-quite hands-on. Additionally, the gov't usually has decent funding for training programs and similar expenses that a contractor has difficulty supporting.
The last item: it's often very dependent on the specific employee, the specifc office, and even the work that is a high priority that year for that office.
Hope this makes sense; if not, I can try to clarify it "next time".