Miracles are God's way of meeting the needs of those who seek Him by faith. Miracles will sometimes be used to demonstrate God's ability so that people may believe (Mark 2:10-11), but they will not happen to skeptics who are tempting God (Luke 4:9-12).
A miracle is a supernatural intervention of God's power over natural law.
Healings occur within the boundaries of natural law while miracles are not limited to natural law.
A person with a high fever who receives prayer and then begins to recover is experiencing a healing. The Lord intervened but in the natural. The virus, infection, or whatever was rebuked left the body (Luke 4:39) and then the natural healing process that the Lord built into all of us took over.
When something totally supernatural happens, that's a miracle. When Jesus re-attached the servant's ear and it was instantly whole after Peter cut it off (Luke 22:51), demonstrated a miracle. It was also a healing, but it was a miraculous healing.
Feeding the five thousand (Matthew 14:19-20), walking on the water (Matthew 14:25), transporting a ship and all aboard to the other side of the sea (John 6:21) all these are miracles. Miracles are usually instantaneous whereas healings are mostly gradual.
The Lord has never had anyone who was qualified working for Him. It's by God's grace that He uses any of us. Anyone who claims to be used of God because of his great holiness is either deceived or a deceiver. It's our faith in the grace of God that allows miracles to flow through us.