烦恼 can be used as a noun. It refers someone/something bothering someone.
The person/thing "bothers" you, not necessarily make you "worried".
他是我的烦恼，因为我整天想他，无法自拔。(He's my 烦恼, because I think of him all day long. I can't control myself.)
他是我的担心，因为我整天想他，无法自拔。(He's my worry, because I think of him all day long...)
外面的噪音令我烦恼。 (The noise outside bothers me.)
外面的噪音令我担心。(The noise outside makes me worried.) Why? Worry what?
In a meeting, you bring up your concern to be discussed, using 担心, not necessarily 烦恼.
我担心的是他无法完成任务。(I'm worried that...)
我烦恼的是他无法完成任务。(I'm bothered by that...) You don't need to be bothered unless some trouble has been made.
They can both be verbs, but:
烦恼某事/某人 rare, often intransitive
我担心他，怕他出事。(I worry about him. I'm afraid of something bad happens to him.)
担心 can be colloquial.
烦恼 is often more literary.
Colloquial (using 烦 instead of 烦恼): 你烦什么呢？别烦了！
Song title: 《你在烦恼什么》