What is the difference between "anywhere" and "other than congested areas" in regard to minimum safe altitude?
The difference is subtle, but here it is:

anywhere means anywhere: At all times you need to operate at an altitude where, should you lose power, you can put your aircraft on the ground without "undue" hazard to anyone on the ground.
Ignoring all airspace considerations, say you're operating over a city (a congested area), and keeping to the bare minimum requirement for altitude and distance from obstacles. Your engine now fails, and your only "landing" options all involve slamming nose-first into the side of a building. You were not operating "at an altitude allowing, if a power unit fails, an emergency landing without undue hazard to persons or property on the surface".
Since we all want to be able to make safe emergency landings this rule is largely common sense with the force of regulation.

other than congested areas is stuff like farmland or open beaches: you COULD operate much lower and closer to obstacles here and still be within the rules above (you're 300 feet AGL over an open field with a barn 300 feet to your right -- The engine dies, you set it down in the field, and everything is cool), but the FAA (and Farmer Brown in the barn milking Bessie) would prefer you not be that close to the barn in the event of say a flight control malfunction.  Most pilots also wouldn't want to be that close to the barn at that low an altitude (unless engaging in an approved aerobatic demonstration with appropriate waivers) anyway, so it's usually a non-issue.

That leaves congested areas - cities, suburbs, crowded beaches, etc - where the FAA bumps up the minimum obstacle clearances / altitudes for the comfort and safety of folks on the ground.
One could contrive a scenario where you could safely fly at 1500AGL next to a 2000-foot building (like a long, clear, mostly-empty park adjacent to it that you could fly over &amp; use for the "anywhere" emergency landing requirement), but again if something else went wrong you might go into the side of the building. As there are more people in "congested areas" to be freaked out by an airplane flying at a safe-but-rather-close distance, and more people who could be hurt by an unexpected departure from controlled flight, the margins are increased to provide a higher minimum level of safety.
As with Bessie in the barn, most pilots probably don't want to be this close to buildings full of people (and the associate mechanical turbulence), and while circling your friends house in the suburbs is fun it doesn't have to be done low enough to pick out the individual bricks in their driveway and you really should be high enough to land on their street safely if your engine quits, which these minimums will likely allow.