I wouldn't say that a human is "likely" to get sick by drinking from a puddle, I'd say "at some risk". It isn't desperately dangerous, although don't take that as a recommendation. There are many infections you can get from water (bacterial, viral, amoebal, and I presume fungal). You don't want to risk them, but probably won't actually encounter them in that puddle in your garden you see your dog drinking from. If the water is stagnant then the list of things in it that might hurt you is longer than if it's flowing. If you can avoid the mud, and anything that's been standing long enough to grow algae, fresh rainwater is a pretty good bet as these things go.
The same is true of other species, but the risk per drink is lower, since they drink it more and develop resistance to more of the things on the list. Humans are no different. For example some areas tend to have Hepatitis A in the tap water. The local population is immune after they've been infected once, and most childhood infections cause no symptoms, so it's not a problem for the locals. But to an unvaccinated visitor, the water's not safe. Hep A normally won't kill you, but is still well worth avoiding since it will make you sick for several weeks.
So basically, humans drink clean water because we can, not because we absolutely need to.
The lifespan of humans living "in the wild" (that is, populations prior to or independent of human agriculture) is pretty well agreed to be rather less than that of typical modern lifestyles. Water-borne disease is only one of many things responsible for that. The expected lifespan of a person used to purified water and so on, dropped into "the wild", is even lower again.
The "clean-water" problem in developing countries is primarily one of high population, relative to what the same area could possibly support in the absence of agriculture and other technology. The biggest risk of water-borne disease comes from contamination by human waste (although there are other sources, for example rats can contaminate water with Weil's disease). A small group of humans living in a large area doesn't have to worry about human contamination, they just drink upstream of where they defecate. This strategy doesn't help much once the population is dense enough that someone else is doing the same thing 20 metres away from you.
And of course, human agriculture also supports a much higher rat population than would otherwise be present (they're omnivores too, so they'll eat pretty much any food humans do). I suspect most human disease can be blamed on agriculture one way or another: if not on the animals then on the close-quarters living. If the population of humans was closer to what it was in the palaeolithic (a few thousand or tens of thousands of people in Europe), and was used to drinking from natural sources, then the water would be mostly (not entirely) safe.
So it's not so much that humans need perfectly clean water, it's that humans have figured out how to live in populations so large that we tend to make all the nearby water particularly dirty. Other species aren't completely unaffected by this, but they are less affected because our waste is specifically contaminating the water with diseases that affect us.