What are the statistical implications of doubling damage on crit instead of doubling the dice rolled?

I’ve been playing with a group that frequently allows players to double the value of dice rolled for crits (and other things) rather than rolling double the dice. Example, someone crits with 2d6 and rolls for 8 damage, which they then double for 16 total crit damage, rather than rolling 4d6.

This mostly just bugs me on principle, but I was curious what doubling the value rather than doubling the dice does mathematically. Does it actually make a difference? Is there a greater chance to hit extreme ends of the range of values (low and high)? Does the amount and type of dice create greater inconsistencies between the two scenarios?