There will probably be more people in general. Not dying of infection from these injuries probably implies that the magic will help take care of most diseases, which is often what killed people in the first place.
I'd be willing to bet that fewer women would die in childbirth. A quick jaunt across the internet (full disclosure: to a not necessarily reliable source, it's a blog) says that in medieval times, 1 in 50 women would die in childbirth. If people were generally healthier and healed faster, childbirth might be easier on the body. There might not be as many midwives. Infant mortality would probably be better, too, and more people would survive to adulthood.
Assuming I'm right about the fewer diseases, cities would be bigger, too. The Ghost Map is a semi-interesting high school reading assignment that explains the constant ebb and flow of population and sanitation. Right now, we can have big cities like New York because modern sanitation/health practices support populations that big. But London of the Victorian era – and certainly of the medieval era – just couldn't handle that large of a population. See: The Black Death… yes, yes, it's Wikipedia, but Europe was overpopulated before the Black Death. If you have a world where people are just generally sturdier, you can have bigger cities with less sanitation.
These cities might actually creep into your narrative. They might be hellish, crowded, and desperately lacking in infrastructure. Overpopulation could be a big issue. Imagine New York. Now Imagine New York without sewers or showers. (Now give everyone a -150 to their Constitution and make them roll a 20 or throw up immediately, because that is going to smell.)
Ah, yes. And speaking of stomachs, overpopulation may cause famine. This is an interesting article on the impact of overpopulation on modern society. Now add medieval agriculture and you're going to have some problems. One way this can manifest is by having food be overpriced whenever any of your players try to buy it.
As far as politics... well, political demography is actually a field of study. You might find some nations trying to implement one child or two child policies in order to cope with the fact that agriculture can't catch up. Or, given that these people have magic, you may find agriculture/technology making leaps and bounds more quickly than we did on Earth. Fewer fatalities means greater life expectancy, and thus a longer time to study, specialize, and innovate. You also may want to glance over this book, which basically explains how modern population growth affects politics. Essentially, more people means less representation. As a result, you may find early forms of socialism emerging in your country's politics... or it'll still be feudal.
EDIT: Two more things. People probably wouldn't need healers as often, but when they are needed, they'll be really needed badly. This might result in having to pay healers more, although they'll get paid less often. If you have a healer or some sort of priest in your adventuring party, they might be affected by this. Maybe a random encounter where someone's kid has a life threatening injury and their parent promises to pay a lot of gold?
Finally, sports. Extreme sports would be played much more often. An injury would be an inconvenience, not something that would permanently put you out of the game. Gladiator games and arena type sports might be more of a thing. And then, of course, culturally, these people probably wouldn't notice/care about pain to the extent that people here on Earth do. Rough-housing might be more common. It doesn't matter if you break your friend's nose/arm, they'll be fine tomorrow/next week, and they were being annoying. Physical injuries would be thought about less often. Truly severe ones would be profound.