My thoughts are that they're two very different movements.
What do you mean you find it easier to balance with a leg behind you? Do you mean something like the so called Bulgarian Split Squat?
If so, here are my thoughts, I'm no expert but have done both of these, weighed as well.
In the BSS your other leg assists in getting up. Yes, you push with it, with it's hip flexors, wether you convince yourself you only have it back there "just for balance" or not. Once you weigh the movement, that leg helps in getting up. In this one it's also much easier to arch the lower back and I had to tighten my midsection hard to prevent this. It also works a lot of the glute-ham tie in... lots of hamstring in my experience. Not that much Quad to be honest. Glute/ham mostly. You can use a barbell or DB's for these, I prefer DB's.
In the Pistol, for whatever reason, I never felt my hamstrings doing much work. Maybe, just maybe, because the Leg in front makes the lower back round quite a lot, and just like some coaches explain for the back squat, when you round at the bottom the hamstrings can't engage very well, right? ... so Pistols are more Glute/Quad for me. And by Quad I mean specifically the VMO or tear drop or inner quad muscle. The leg that's in front, at first, nearly cramped my hip flexor from holding it straight in front. But eventually you can hold it there no problem. I weighed my pistols by holding dumbells or plates in the "rack" possition (like a goble squat or front squat). Yes, balance is an issue, but that's one the good things about the pistols, develops balance.
My two cents but keep in mind I'm not an expert and not qualified to give advise, but I was bored.