Punting on third down

Assuming there's no attempt to disguise that it'll be a punt, any more than there would be so on 4th down....

It was done more when the percentages favored it more.  What tips the percentages in favor?

One would be if you saw a need to give the other team a little more doubt that it would be a kick, to distract them from getting a good rush or especially a good return.  This effect isn't as great as it would be for a quick kick, but if the other team's been especially good at playing against punts, forcing them into a "punt safe" defense might help.  Of course that's only if you expected to be punting on the next down anyway, which leads to the next point....

Your offense stinks, but so does the other team's.  (Or your defenses are both great.)  Let's say time is becoming a serious factor, you're trailing by a TD or less, and you're in long yardage.  In that case, you might be able to get a first down sooner by an exchange of punts.  You're still probably going to lose, but you may have a better chance to pull out a scoring drive by starting fresh after making the other team punt (or forcing a turnover) than the chance of your converting this 3rd down and long.

Another would be if teams have had a lot of trouble holding onto the ball.  In that case, you might like to set the other team up for failure in worse field position for them, better for you, by punting on an early down.  Maybe 3rd, 2nd, or even 1st.  What this means is that it's been a game where possession of the ball hasn't counted for much, because you don't know who's going to be snapping it next, and maybe you'll get lucky in the other team's territory.  It's about winning ugly.

I forgot about the reason Bignose gave.  That was the reason for most of the 3rd down punts from punt formation I've seen in this century.

If you're looking at game film from 100 years ago, you'll get a distorted impression because of a certain rule that was in effect for a little over a decade early in the 20th Century.  Remember the XFL of 20 years ago, and how if a scrimmage kick hit the ground at least 25 yards beyond the line, it was either team's to recover and keep?  That was a revival of this old rule.  (Originally it was any distance, then after a couple years they required it be 20+ yards.)  Looking at a YouTube of this old game, each team would come out in deep punt formation on most downs in their own territory, forcing the defense to keep people back against the threat.  It was the rules committee's idea of opening up things for the offense.  (Oddly enough, the quick kick was practically outlawed for most of that period, requiring scrimmage kicks, as well as forward passes, to originate from at least 5 yards behing the line.)

Are you asking about a "quick kick"  (surprise kick, usually by the QB) used to take advantage of no return man?  Or are you asking about a team just lining up in a punt formation on 3rd down with no element of surprise? 

What mostly increases the distance is that the ball rolls because there's nobody to field it.  The distance of the kick in the air is usually somewhat less than a regular punt because of the style of approach to the ball.  On a quick kick instead of 2-3 straight steps into it, you might actually use a rocker step.  Or you might be stepping sideways into it and using a scissor technique, resulting in a relatively low kick.

If your quick kick is the low arc style, then you should not use it to take advantage of a wind at your back, but you could use it to mitigate the effect of a wind in your face.