Believing that being a DM is all about rules would be a great disservice to both you and your players. That's both dry and narrow. The DM isn't always the person at the table who knows the rules best; sometimes people make the jump for creative reasons as well. Maybe no one else in the group wants to run the game. Rules are a part, of course, but there's also preparation, multiple characters, collaboration and moderation that are thrown into the mix. Running games isn't something a rulebook does on it's own!
While rules aren't the end-all be-all, they are an obvious and important piece of running any game. As a player, you need know only what rules apply to your character and whatever environment the game is set in. When you are the DM, not only do you need to know the player's characters well enough to plan for them and the environment, but you also need to know whatever tactics the foes and allies of the players will use. This doesn't mean you need to sit down and study the rulebook front-to-back with pen and paper, though. Pick out just those rules surrounding the characters and whatever you prepare for the next session. Over time, you'll have prepared a diverse amount of encounters and played enough of a variety of characters to have a solid understanding of each rule you or the players are likely to encounter.
Artistically speaking, preparation is the greatest advantage to being the DM. The story your group tells via their characters is a collaborative one, but since you control “the rest of the world,” from the players' perspectives, you've got a lot more say in what types of challenges the group will face. It's your responsibility to either make those encounters yourself, find a prebuilt adventure module or whip them up on the spot. How much health will their enemies have? What strategic advantages will be available? Why does the plot take them to this place? Where will the villain strike next? All of these questions are yours to puzzle out: Create your piece of the session so the players can input theirs.
As a player character, you had either one or very few characters in your control. Whether you were a bossy thief, disgruntled arcanist or a solemn priest was up to you. In fact, that's all you had to concern yourself with; Other players had their characters and then the DM covered the rest. Covering the rest is simultaneously great fun and hard work. You've got to keep continuity – if the bartender speaks with a funny accent and a lisp on one night, the players will come to expect that. If they drop their speech patterns, then why? Each vampire spawn has the potential to be captured and interrogated so you'd better have a quick name ready in case that's what the group does. Portraying the world is most easily done in thinking big but playing small: give life to individual characters as they come up and keep in mind that in any given town, any number of personalities and perspectives will exist.
Both referee and integrator are roles the DM takes in a typical group. Should someone in the group get rude or otherwise out of hand in regards to the game it's up to the DM to stop the game so the issue can be addressed. Rules-lawyers will be on you at any given turn so it's your responsibility to establish with them proper etiquette on when to correct and when not to correct the rule calls you make. In spite of those, the show must go on. Every player decision could have the chance to affect something within the game world: If they assault the mayor, the constable will react with force (usually); Insults are not something the egotistical villain is likely to forget; When the players purchase every sword in town to arm their undead army, the price of swords would probably skyrocket. Whichever elements you take from the players' choices help make the story deeper, fuller and more collaborative. Let your players supply the brushes, you supply the ink and the game world be the canvas. Happy transitioning!