As with "go your own way," march to the beat of your own drum has positive and negative connotations simultaneously. The connotations of each phrase are different, though. No simple, abstract description can fully convey the connotations of either phrase. To understand them, you must follow some of the cultural landmarks that echo in people's minds when they hear each phrase.
First, here is the passage from Walden, by Henry David Thoreau, which is usually considered the origin of the expression:
Why should we be in such desperate haste to succeed, and in such desperate enterprises? If a man does not keep pace with his companions, perhaps it is because he hears a different drummer. Let him step to the music which he hears, however measured or far away. It is not important that he should mature as soon as an apple tree or an oak.
The phrase refers to people marching in formation, their steps synchronized by a drummer whom they all hear. Now imagine that one man is out of step with the rest of the formation—he isn't "keeping pace".
The "formation" that Thoreau is referring to is the common, conventional practice of seeking money, physical comforts, and the respect of most people—which is how most people live. But some people don't care very much about conventional success. Some people feel called to unusual pursuits that most people don't respect and that don't pay much money, like joining the circus or the life of a hobo. These people are "marching to the beat of a different drum" than the "drum" that guides ordinary people.
So, saying that someone marches to the beat of a different drum suggests that they are weird, maybe somewhat goofy, certainly idiosyncratic. It suggests that they probably will not "succeed" by conventional criteria of wealth and public respect (though in rare cases they might become extremely wealthy).
Just as with "Go Your Own Way" (1977), there is a well-known pop song called "Different Drum" (1967)—and the words are, again, those of a person rejecting a boyfriend/girlfriend because they think their lives need to follow different paths. In "Different Drum", the girl mainly needs to maintain her freedom and independence, which would be lost if she stayed with a boyfriend who adores her too much. The rejection in "Go Your Own Way" has a more conventional reason: "You don't care about me enough. So, you will have to do without me, i.e. go your own way."
A description, not a prescription
Another difference between the two phrases is that it makes more sense to say that a person marches to the beat of a different drum, than to tell a person to march to the beat of a different drum. Quirky, unconventional people are not that way by choice. The phrase explains their odd behavior: they march differently because they hear a different drum than the one that most people hear. It doesn't make sense to tell an ordinary person to march to the beat of a different drum. Ordinary people hear the same drum that most people hear. They don't hear the "far away" drum that quirky people hear.
"Go your own way" suggests self-reliance more than it suggests quirkiness. The Frank Sinatra song "I Did It My Way" suggests that even though he chose the hard, individualist path, he still succeeded by the usual standards of wealth and respect—more than most people, perhaps. "Go your own way" makes more sense as advice—and it can even serve as a contemptuous dismissal.