Like the name would suggest the key difference between the two is how they are fed, with the typical question being that of ethics. A cow on a natural diet of grass is more likely to be living an organic lifestyle living off the farmland, an overall more natural life. Especially when compared to corn fed cows, who live in cramped quarters where the profit line is key, the cows are shown to live a far more stressful life when corn fed. It is quite a hard stance to take as everyone’s moral compass does differ, grass fed cows are shown to be happier, but they do take up more environmental space and cause more environmental damage, both do face the same fate of death, so if you really want to make a difference go vegetarian.
It has been shown that grass fed beef is better for you than corn fed beef. As a study by The European Journal of Clinical Nutrition showed grass-fed beef had significantly more omega 3 fatty acids and a higher anti-oxidant content while also being leaner, and therefore healthier than grain-fed beef.
What the majority of people really care about is the difference in taste. The diet of a grass-fed cow is obviously a lot less micro managed so there is far more variety in their diet meaning more variety in the flavour of the beef and is also typically a lot more lean and less fatty.
Corn fed cows all face the same diet so there is less variety in the flavour, but there is a consistency in its fat and texture, as every corn-fed cow will taste extremely similar there is less risk involved, you know what you are going to get. This is all subjective, and most people do carry their own preference.
If you look at the facts above, I would say if you are looking for health benefits then go for grass fed beef. If you are looking at this from an ethical stand point, I do not think there is a clear winner as both cause mass eco damage. Finally, from a taste stand point I would recommend trying both and just choosing a preference. It is important to remember that grass fed beef is typically more expensive so do try think with the facts above if it is truly worth it.