Nobody has presented good evidence that eating meat from grass-fed animals is more healthful than the meat from corn-fed animals. The main health arguments for eating grass-fed meat are its lower fat content and higher content of omega-3 fatty acids (Meat Science, January 2014; 96(1):535-540).
Fat: There is no evidence that the small reduction in fat from eating meat from grass-fed cows, in place of corn-fed cows, offers any health advantage. It's the total fat in a diet, not just the fat in meat, that determines blood test results that check on blood levels of the heart-attack-predicting LDL cholesterol and triglycerides. Reducing dietary fat sometimes lowers LDL, but often raises triglycerides.
Omega-3s: As for meat from grass-fed cows having more omega-3 fatty acids, the difference between the diets of people eating the different meats is inconsequential and irrelevant. The difference between four ounces of grass-fed or corn-fed sirloin is 22 milligrams of omega-3s. The same portion of salmon contains 1,270 milligrams of omega-3s. All animals, including humans, get their omega-3s from the food they eat. Ocean fish have lots of omega-3s because they eat smaller fish or plankton that are rich sources of omega-3s. Fresh-water pond fish usually are low in omega-3s because they eat little or no plankton. Cows get a small amount of omega-3 fats from grass or clover, but they get more when they also eat corn.
The argument that grass-fed meat contains less fat and more omega-3s is probably irrelevant because recent studies show that taking extra omega 3s may not help to prevent heart attacks, and many serious scientists believe that other factors will be found to account for the association between eating meat from mammals and increased risk for cancer, diabetes or heart attacks, such as TMAO or Neu5Gc.
Flavor Preferences and Cost
Most people prefer the flavor of corn-fed meat (Nutr J, 2010;9:10) because an increased amount of fat is usually what makes meat taste good and corn fed meat has more fat (Journal of Dairy Science, 97 :1828–1834). Grass-fed meat is usually more expensive, since it takes more land to feed the cows only grass and it takes longer for cows to reach market weight. However, heavy advertising to persuade people that grass-fed meat is healthier can also influence their taste preferences (Int J of Hosp Man, September 2014;42:137–143) and their willingness to pay higher prices.
You will not prevent heart attacks or diabetes just by choosing meat with less fat. I believe that a healthful diet is low in meats from mammals (fresh or processed), regardless of their fat content. If you want to eat more omega-3 fatty acids, eat ocean fish and a variety of seeds.