Julia Belluz also wondered about this and checked the scientific literature to answer this question.
The first thing she found in the literature was that running indeed is much more likely to lead to injuries. More than half of the people who run, at some point, get injured, while the risk of injury while walking is minimal (around 1% gets injuries). But from her literature scan Belluz could not find out whether running, apart from the injuries, leads to more health benefits. Both running and walking have demonstrable large health benefits.
She talked to some experts who confirmed that both activities have important physical and psychological benefits such as reduced stress, depressive symptoms, and emotional problems. Based on these conversations she concludes that running is a slightly more efficient way to improve health. But there are two caveats. First, as mentioned above, there is an increased chance of injuries, second, extreme running exercise can lead to health damage.
Reflection: it seems that the choice for walking or running is mainly dependent on individual preferences. Running appears to lead to health benefits a bit more fast and efficiently but is also more risky. Then again, there must be things one can do to limit this risk, such as slowly building up your exercise. Anyway, my choice is clear. I am fan of walking. Not only because of the low injury risk but also because of the practical convenience of walking. You can do it anywhere and always. you do not need special clothes or footwear, you can have a conversation while doing it, you can look around you and enjoy nature, etc. Also, I have noticed that a 20 minute walk can have a stimulating effect on your thinking (it increases your creativity). I once read about this effect in a paper and have tried it out while writing my most recent book. Each time I got stuck in the writing process, I went out for a walk. Nearly always I came back with a clear idea about how to proceed.