Sqn Ldr in the RAF, Managing Director in civvy street. Don't know if that qualifies me to make a comment...but in for tuppence.
The biggest change IMHO is one of culture...
Armed Forces may be different to other part of the public sector, but they are still the public sector and, defacto, not motivated by commercial necessity (that would be "profit").
Commercial companies (in a capitalist free market economy/society) are entirely profit driven (let's not digress in a pointless discussion here...if anybody thinks profit is not the ultimate goal of a commercial entity, then they just don't get business...period).
So the leadership goals and priorities (and therefore applied techniques) of public sector and military leaders can look profoundly different. Case in point...
You have an underachiever in the public sector...leadership priority...make them better.
You have an underachiever in the commercial sector...leadership priority...sack em.
Too harsh...check out Karren Brady, CBE, Baroness, Managing Director of BCFC at 23. To quote David Gold "She's a sacker". She is also an incredibly effective leader (and I venture she would be in the military too)
But ultimately therein lies the common ground - Leadership is Leadership, and more often than not it's about taking the tough decisions, without fear or favour, that nobody else wants to take.
That's why tons of people talk bull about leaders when they themselves can't hack it.
As a leader you are invariably responsible to an organisation for the delivery of a task - quite often as the expense of the individual to the benefit of the larger group/crowd.
I did a fair bit of leadership in 27 years in the military...team leader, crew captain, detachment commander, Sqn Exec. I was well trained for the task and in the main everybody I led was perfectly capable of replacing me.
It was a far, far harder gig trying to protect the jobs of two dozen people in a recession. And "letting go" of two married guys with families to support was one of the worst days of my life. It was also IMHO, a true act of leadership.
PS Karren Brady was just an easy example to make a simple point - I can't stand the Apprentice or her on it either!!