I've wondered for a long time what the difference is between praying for healing versus receiving a priesthood blessing.  Or in other words, why would one seek a priesthood blessing over just praying to get better?

While studying this topic, I learned a few things that I thought were significant.  First is that we believe in trying all remedies for an illness.  Prayer alone without any action isn't what the Lord wants us to do... just the same as we don't pray for a harvest when we don't do any work to put seed into the ground.

Second, we need to remember that doctors, medicine, etc. doesn't heal us, they just create the ideal environment for healing to occur and our body, designed by the Lord and surely under his care, heals itself.

Third "a ha" is that the Lord has ordained a sacred ordinance for the healing of the sick. It involves anointing with consecrated oil (symbolic of blessings being poured out upon us) and sealing the anointing.  The words of the prayer don't matter as much as the anointing and sealing, as that's the method through which the Lord has ordained healing.  This ordinance is explained a little bit in modern revelation as well as in James 5.

Fourth, and probably most important, is that faith of the recipient of the blessing is very important.  Even when Jesus performed miracle after miracle, he said "Thy faith hath made thee whole."  We need to have faith that the Lord can heal us.

Lastly, we should remember to give God the glory for our healings.  It's going to be his will whether or not we'll get better or die, and it does us no good (and it's not true faith) if we demand a blessing from the Lord to be answered in the way we want.  Remember to give God the glory for our healings, not the person who gave the blessing.

See also: Elder Oaks, April 2010 General Conference; Spencer W Kimball, October 1981 New Era.