Volunteer George Brown, 68, from High Etherley, County Durham, was killed on Monday afternoon on the A688 between Staindrop and Bishop Auckland.
He was driving to collect patients on behalf of Butterwick Hospice in Bishop Auckland when the tree fell on to the ambulance.
Passenger Peter Foster, 68, from Crook, County Durham, also a volunteer driver, suffered non life-threatening injuries and was being treated at Darlington Memorial Hospital.
On Tuesday, hospice bosses paid tribute to Mr Brown, who leaves a wife, Mary, a son, a daughter and three grandchildren.
Graham Leggatt-Chidgey, chief executive of the Butterwick, said in a statement: "This is a terrible tragedy for George's family, friends and colleagues. We have lost a selfless and loyal member of our team at the Butterwick and his family has lost a husband, father and grandfather.
"George has been a volunteer driver since 1999 and was greatly respected by everyone who knew him. There is a deep sense of shock throughout the hospice team at this awful loss. We send his wife, Mary, and his family our deepest sympathy and our prayers and will support them in any way we can."
Thousands were left without power on Monday night as the strong winds caused damage to buildings and disrupted travel.
Firefighters in West Yorkshire said they were called to a number of other weather-related incidents on Monday including two in which telegraph poles came down, setting fields on fire. A spokeswoman for Durham and Darlington Fire and Rescue Service said no-one was injured in the incident.
Steve Ellison, a forecaster at MeteoGroup, the weather division of the Press Association, said conditions had eased on Tuesday although winds remained strong. "We're looking at winds of just over 50mph, possibly reaching 60mph in the most exposed areas," he said.