What's the difference between these 3?

Asset management in the traditional sense of the word means long-only, no leverage, public market assets that are indexed to a benchmark. A household name in 'asset managment' would be PIMCO or Blackrock. A hedge fund tends to be managed to an absolute return mandate which means they are intending to achieve a target risk/return profile regardless of how markets are doing. Hedge funds are more opportunistic, go-anywhere, can use significant leverage and invest in illiquid opportunities. This goes into the idea of 'alpha' - asset management shops intend to achieve 50-100 bps of alpha whereas hedge funds will try and hedge out market exposure and make their entire return 'alpha.' The fees and incentives are very different - asset management = asset based fees, aggregate assets, don't screw up. Hedge fund = management fee + carry, raise a couple hundred million and then hit home runs so you get carry.

Not sure what you are trying to get at regarding the AM that invests in private businesses vs private equity firm, but here's my best stab at it: large asset management firms have smaller arms that may take stakes in private businesses. i.e., Fidelity has been mentioned as an investor in uber and a number of other tech firms. I'm not sure if the proceeds are coming from a specific fund, but the thought here is probably that there's some money to be made by getting in on late stage to-be public companies a bit earlier. To your point no operations control, and intention is probably a long term hold. Private equity is raising institutional capital to buy controlling stakes in firms with the intention of flipping them for a profit somewhere down the line (5-7 years). These tend to be smaller mid-market companies.

Hope this helps.