FINALLY! a great question (since we are talking about Biblical prayer):
Matt 17:14 And when they were come to the multitude, there came to him a certain man, kneeling down to him, and saying,
15 Lord, have mercy on my son: for he is lunatick, and sore vexed: for ofttimes he falleth into the fire, and oft into the water.
16 And I brought him to thy disciples, and they could not cure him.
17 Then Jesus answered and said, O faithless and perverse generation, how long shall I be with you? how long shall I suffer you? bring him hither to me.
18 And Jesus rebuked the devil; and he departed out of him: and the child was cured from that very hour.
19 Then came the disciples to Jesus apart, and said, Why could not we cast him out?
20 And Jesus said unto them, Because of your unbelief:
The difference is answered prayers vs. not answered prayers. Analogy: the difference is between using aspirin when you needed an antibiotic--one produces the desired results and one doesn't.