An actively managed fund – more commonly referred to as a mutual fund – has a higher risk versus reward value, is much less passive and gives greater control to an individual investor than a simple index fund does.
Index funds are passive. They mirror the market as a whole as closely as possible, investing as widely as they can to avoid massive plummets in value. Ordinarily, this means that an index fund is fairly safe, but if the market itself is heavily involved with stocks that are unsafe or just inflated in value, then the inevitable crash destroys the value of your index fund. A managed fund, on the other hand, can identify those unsafe funds and either avoid them or attempt to live the "Buy low, sell high" mantra.
As an example, the Nasdaq index fund was heavily invested in the dot-com bubble of 1997-2000 by virtue of its focus on the technology sector. While the index itself collapsed in value afterwards, individual managers had the chance to avoid the whole situation. Several played it savvy and either sold early or hung onto the stock of companies that survived such as Amazon.com, which went from $107 to $7 a share but rebounded dramatically over the following decade. As of 2015, Amazon hasn't been below $200 a share since 2012.
Conversely, managers often look for stock of companies that are undervalued by the market as a whole, picking them up before the general index twigs to their real value. This is called value investing, and it is the primary draw when investing in a managed fund.
Another downside to indexes is that investors are involved with and supporting all the companies on that index. If an investor dislikes a company for moral or personal reasons but that company is on his index, he has no way of removing his money from that company without exiting the index fund entirely. With a managed fund, on the other hand, all it takes is a simple call to the manager stating that he does not want to have money involved with that company.
Risk Vs. Reward
Investment is comparable to legalized gambling, but instead of putting money toward random balls bouncing across a wheel, an investor is putting money toward which companies will do well in the future and away from those which will not. Index funds are like betting on red every time; it's safe, which is both its upside and downside. Choosing one's own investments or leaving them to a trader is risky because it involves the human element, but it's also where the big rewards come from. Make the choice between indexed and managed funds knowing that difference.