Sapphire crystal is harder and more scratch-resistant, while mineral crystal is cheaper and more resistant to shattering on a hard impact. Both do the same basic job of protecting the dial, but they trade off in almost opposite ways, so which one you want depends on what kind of damage you're actually trying to avoid on a daily basis.
Sapphire barely scratches, but it can crack
Sapphire crystal is synthetic corundum, the second hardest natural material after diamond, which means everyday scratches from keys, tools, or rough surfaces essentially don't happen. A watch with a sapphire crystal can look brand new years later even with regular wear, which is a big part of why it's the standard on higher-end watches. The tradeoff is brittleness. Because sapphire is so hard, it's also less flexible, and a hard direct impact, like dropping the watch on concrete at the wrong angle, can crack or shatter it rather than just scuffing the surface the way mineral glass would.
Mineral crystal takes a hit better than it takes a scratch
Mineral crystal is treated glass, softer than sapphire, which means it will pick up fine scratches over time from normal daily wear. Those scratches are usually shallow and can often be polished out with a bit of toothpaste or a dedicated polishing compound, but they will show up eventually on any watch worn regularly. What mineral crystal gives up in scratch resistance, it gains in impact resistance, since the softer material flexes slightly rather than shattering under a hard knock.
For an everyday, affordable watch you plan to wear hard, mineral crystal is a perfectly reasonable choice and part of why it shows up on so many field watches and vintage-style pieces. For a watch you want to look pristine indefinitely with minimal care, sapphire is worth the extra cost, especially if it's a watch you plan to keep for a long time.