The church of Jesus Christ of Latter-day Saints make a distinction between the two. There are at least two powers that can be used in healing:
We know that the prayer of faith, uttered alone or in our homes or places of worship, can be effective to heal the sick.1
... healing blessings involving the power of the priesthood. ...
Many scriptures teach that the servants of the Lord “shall lay hands on the sick, and they shall recover” (Mark 16:18). Miracles happen when the authority of the priesthood is used to bless the sick.1
In regards to both:
... we must always remember that faith and the healing power of the priesthood cannot produce a result contrary to the will of Him whose priesthood it is.1
1 Healing the Sick, Dallin H Oaks
See also Healing, History
A small note, the gift of healing is noted as a gift of the spirit, but those without this gift are not excluded from healing. Askgramps (an unofficial LDS Q&A site) provides some clarification on the distinction:
...if someone has been blessed with the gift to heal, will their blessing have more power than someone who hasn’t? If I may, I would like to change one word and that word in this question is “powerful.” A powerful blessing is a blessing that is given by the Spirit from a worthy priesthood holder, no matter what spiritual gifts this person has. The power of a blessing is based upon the individual worthiness of the priesthood holder, not their gifts. With that said, I would like to change the word from “powerful” to “efficacy.”
If a blessing has more efficacy it will be a blessing that is more likely to produce the intended hope through our faith. Therefore, a blessing that is received by a person who has been blessed with the gift to heal will more likely produce the intended result with the caveat — according to the will and pleasure of the Lord.
Will a blessing then coming from a priesthood holder who has the gift to heal be more powerful then? No.
Will a blessing then coming from a priesthood holder who has the gift to heal have more efficacy? Yes, it is more likely this blessing will produce the intended results of our hope through our faith in Christ, assuming it is the Lord’s will.