A "snake" draft is a draft in which the team with the first overall pick gets the last pick of the second round, then goes first in the third round and last in the fourth, etc. It narrows the advantage of going first since you get that advantage in only half of rounds and receiving a partially-offsetting disadvantage in the even numbered-rounds. It is common in fantasy sports and seems to work well there, although most fantasy leagues start the season with everyone on equal footing in personnel terms (everyone has zero players) unlike in real professional sports where teams that did well in the previous season will likely have stronger rosters than those that did poorly.
The purpose of the snake draft is to try and not give an advantage to any team drafting, under the assumption that having the 1 and 16 picks in an 8 team league is about the same value as 8 and 9. This might not be so, but it is probably as close to favoring nobody as you can get.
The usual purpose of a league draft is the opposite - to give an advantage to poor teams to provide better competitive balance for the league going forward. The NBA mitigates the advantage a little with the lottery, so that there isn't as large an advantage in tanking the season to have the worst record to get the #1 pick overall.
So leagues don't use snake drafts because the goals are different.