The tree, near Brook House, Chetwynd End, was more than 50 years old.
The trunk, which measured 1.3 metres in diameter, was removed in December 2018 and cut up into logs.
The willow was on land owned by Sanctuary Housing which used the services of tree surgeons CTS Environmental Management Ltd.
Magistrates sitting in Telford heard that a member of the public reported the felling to Telford & Wrekin Council which had not received statutory prior notice of the work.
The court was told that each defendant thought the other had notified the council.
Telford & Wrekin Council said had notice been received, a tree preservation order would have been made as the tree was a prominent feature and not diseased.
CTS Environmental Management Ltd of Kingsbury Road, Erdington, Birmingham, pleaded guilty to cutting down the weeping willow contrary to the Town and Planning Act 1990.
It was fined £1,000 and ordered to pay costs of £620 and a victim surcharge of £100.
Sanctuary Housing, of Castle Street, Worcester, admitted to causing the tree to be cut down, contrary to the same law.
It was fined £2,000 and ordered to pay £660 in costs and a £170 victim surcharge.
Speaking after the case, Councillor Richard Overton, Telford & Wrekin Council’s cabinet member for enforcement said: “Trees are an important part of the eco-system within the urban environment.
“They provide oxygen, store carbon and are a cornerstone within the food chain for the ecology and the biodiversity of the area.
“Within conservation areas most trees over 75mm in diameter are protected by the Town and Country Planning Act.
“It means, any work done to the tree, such as felling or pruning will need an application to the local authority.
“Any work that has not been applied for may be liable to prosecution. If anyone is unsure then I would urge them to check with the council.”
Visit telford.gov.uk/info/20170/planning_applications_and_guidance/108/protected_treesfor more information about protected trees.