According to Dictionary.com, implement means to put into effect according to or by means of a definite plan or procedure. So basically, implement means to begin, to put into action, to set on, to start a course of action.
According to the same source, execute means to perform or accomplish something, as an assigned task. So the word execute means to see through to completion.
In your situation, the UNEP agency would be the agency responsible for starting projects while the Department of Environment agency would be the agency responsible for completing projects.