What is the relationship of the penalty between quickly using the hint button and spending 15 seconds..

Definitely choose the hint button. You'll lose maybe 10,000 points. But if you take >7 seconds to find an object, you will lose the bonus bar, which could cost you a lot more points, depending how high your multiplier was and if you have to restart the bonus bar. Either way you lose the "Perfect" score, which you earn for no hints, no misses, no lapse of the bonus bar.