In addition, team teaching is a participle phrase wherein 'team' functions as an adverb (i.e, teach together. Teach how? Together). The phrase 'team teaching'--because it's a participle--functions as (a) part of a verb (e.g., They teach together ~ They team teach, They are team teaching right now), (b) an adjective (e.g., They are a team teaching machine), and (c) as a gerund (e.g., Team teaching is fun!).
teaching team , on the other hand, does not refer to an action. It's a noun phrase. It contains the adjective 'teaching' and the noun 'team'. 'teaching team' refers to a thing and it means, a team that teaches.
Question: What kind of team?
Answer: The teaching kind.
We are team teaching right now.
(We are teaching together as a team right now.)
We will team teach tomorrow. (Future)
(We will teach as a team tomorrow.)
We team taught last week. (Past)
(We taught as a team last week.)
We are a teaching team.
(We are a team, one that teaches together.)