(edit - Whoops! Completely missed that this question is about a dividend paid in stock, rather than cash)
These are completely different things:
Dividend: The company takes money out of their bank account and gives that money to each shareholder. No shares change hands. All else being equal, this usually causes the shares to drop a bit (roughly equal to the amount of the dividend), as the company now has less money.
Stock split: The number of shares doubles universally. Everyone who had one share, now has two. That's it. All else equal, the share price roughly halves (in a 2:1 split), as there are twice as many shares and presumably the total value (market cap) of the company has not changed. In practice there is some idea that a cheaper share price might induce more buyers and slightly elevated the price after halving it, but no guarantee of that.