My profile page is saying I awarded 300 points for a bounty.
While this page is saying that I awarded 350 points for the bounty.
You did indeed award 300 of your own reputation, so the number in your profile is correct.
The answerer, however, did indeed receive 350 reputation for this, because this bounty happend in May of 2010, which was before the changes to the bounty system. Back then, the system threw in an additional 50 rep to bounties (this is no longer the case).
So both values are correct.