Grass-fed beef is just a bit tougher and has more texture than grain-fed. But to me it has a lot more flavor. I don't think it necessarily has to do with the fat content, as this is true of the hangar, skirt, and ribeye that I get from grass-fed cows.
Grass-fed cows tend to be pastured, which means they walk around more, which means they use their muscles more, which is what makes them tougher and have more flavor. This is also what makes the meat more nutritious.
I think one of the motivations for grain-fed beef (besides cost) is that it makes the meat more tender, partly because you're able to feed the cows without them walking around (which is bizarre when you stop to think about it). However it is exactly this that makes the meat less nutritious, and with much less flavor.
In some places you can find "grass finished" or "grain finished" beef, which means the cows were raised on grain or grass, and were given grass or grain for a while before slaughter. This is supposed to provide some qualities of both grain and grass fed, i.e. tender meat that is grass fed. However, for me this also eliminates a large reason to eat grass-fed beef to begin with, which is to eat beef that is free-range and fully grass fed.
Many say that beef is "supposed" to be somewhat chewy and fully flavored, and I think this kind of beef is most nutritious. You might look at cooking techniques that make the beef more tender, such as tenderizing it with a mallet or other device. You can also braise or slow-roast many cuts until they are falling-apart tender. One of my favorite dishes is stuffed braised flank steak, which cuts with a fork. Generally meat is tenderest when cooked briefly at high heat (such as on the grill), or at low heat for a long time (braising).