Power
Unlike a hedge fund manager, you can't short stocks, so your power is different. It's more about the power to do good than the power to do evil. See Google for details.
You'll be buying and selling stocks like nobody's business, and that should make you feel pretty powerful. The real power, however, comes from bulk. Mutual funds tend to be quite a bit larger and coalesce into large buying groups.
Let's say a contentious CEO is spending money on jets for herself instead of buying back stock, like shareholders want. The fund managers usually have a friendly (which means it isn't) with the CEO and explain that the new board elections are coming up—either she behaved or the funds elect their own board candidate, and the CEO, instead of having a nice career, gets two weeks' notice and a free subscription to Monster.com, which she will desperately need.
They behave. Or they die. Yeah, it's a rush.
And access moves all around the food chain. Own twelve percent of a company's stock and you really want to meet with the head of marketing? Sales? Technology? Distribution? Product? Likely very doable. The investor relations person will likely be happy to pour coffee for any and all meetings that you'd want to do if it'll keep you long the stock. It's a wonderful chance to get to know a business better than anyone else in the...business. Do it.