Dedicated to the Lord Jesus Christ Who loved us, gave Himself for us, and washed us from our sins in His own Blood; and to the Church, which is His body.

Prayer and fasting do not drive out certain demons . If the name of Jesus and faith in His name can't, then fasting and prayer won't either. Jesus is saying that fasting and prayer are the only ways of casting out this type of unbelief (Mark 9:28-29).
An unbelief that comes as a result of ignorance can be done away with by receiving the truth of God's Word (Romans 10:17; 2 Peter 1:4). The unbelief that hindered the disciples in this case was a "natural" type of unbelief. They had been taught all of their lives to believe what their five senses told them. They were simply dominated by the natural more than by God's supernatural (God's Word). The only way to overcome this unbelief that comes through our senses is to deny our senses through prayer and fasting.
Fasting accomplishes many things. One of the greatest benefits of fasting is that through denying the lusts of the flesh, the spirit man grows. Fasting was always used as a means of seeking God to the exclusion of all else. Fasting does not cast out demons but rather, it casts out unbelief. Fasting is beneficial in every area of the Christian life. 
The virtue of a fast is in humbling ourselves through self denial (Psalm 35:13; 69:10), and that can be accomplished through ways other than total abstinence. Partial fasts can be beneficial, as well as fasts of our time or pleasures. Because appetite for food is one of man's strongest drives, fasting from food seems to get the job done the quickest. Fasting should be a much more important part of our seeking God.