The question is a complete no-brainer. Jack is, hands down, the greatest professional golfer that ever walked a course. Anybody who plays the game knows that great golf is played between the ears and Jack's complete unflappability would disarm Tiger. Did you ever see Jack Nicklaus blow up on a course? Tiger has more natural talent and all the tools but he hasn't got the head to beat Jack.
As far as greater competition at the time, how many multiple major championship winners has Tiger had to compete against.( I settled for 4 or more to have real presence on my list.) Ernie Els at 4 and Phil Mickelson at 5. During his playing years Jack competed for majors against Gary Player at 9, Tom Watson at 8, Arnold Palmer at 7, Lee Trevino at 6; Seve Ballesteros at 5, and Ray Floyd at 4. To put it more succinctly, Tiger's big competitors 9, Jack's 39. I think that suggests that Jack had the tougher "row to hoe" when it came to competing for major titles.
Tiger is a greatest golfer of his time but he's no Jack Nicklaus.