This post was largely inspired by a post in /r/magicTCG recently about a Heathstone player attending a MTG event for the first time. There's a lot of overlap between the two games, but MTG is a more complex game, and there are significant differences between the two that a Hearthstone player should understand before jumping feet first into MTG. I've played both games (MTG much more) but I'm by no means an expert. However, this will be a somewhat in-depth guide to the major changes a Heathstone player should expect if they want to try MTG.
If this makes it seem like the game is exceedingly complex, it really isn't. There are lots of interactions, but the rules themselves are straightforward, and the things each card does is written right on the card. If you really want to learn, go to your Local Magic: The Gathering Game Store (LGS) or pick up Magic Duels: Origins for free on Steam or the new consoles.
Something to note: if I were to talk about every exception to every point I'd be here forever. So assume that there are exceptions, and unless I talk about them they're usually not relevant.
Mana Both games have a resource called Mana which you use each turn to cast spells, and replenishes when your turn starts again. However, in Hearthstone mana is given to you without expending other resources, and except for certain Shaman or Druid decks, you always get an additional mana to use each turn, up to 10. In Magic, players get mana by playing and using Lands. By default, you can play one land from your hand per turn. Each turn, you may tap a land to produce one mana of the appropriate color; plains produce white mana, forests make green, mountains make red, swamps make black and islands make blue. You use mana to pay the cost of a spell, which is located at the top left corner. For example, Guardian of Tazeem requires 5 mana total to cast. Two mana must be blue (the water drops), and three can be mana of any color. It's important to note that unlike Hearthstone you're not guaranteed to get a new land to play each turn, and lands also take up space in your deck. Most decks run about 40% lands.
Color In Hearthstone, you have 10 classes, each of which gets access to exclusive cards, and a bunch of other cards that can be used by any class. In Magic, you have 5 different colors; White, Blue, Black, Red, and Green, often shortened to WUBRG (U is for blue). Decks may play as many colors as they wish and in any combination, so any deck can play any format-legal card. Each color has a unique role, along with strengths and weaknesses. This is known as the color pie. You can learn everything you'd want to know about color and the color pie at MTGSalvation's page. It's important to note that while the meta changes from set to set, no one color is inherently stronger than another. It's simply that each color answers the question "how do I win?" differently. Also, while most cards have one color, some have more than one, and some have none.
Card Types In Hearthstone, you have three types of cards: Creatures, Spells, and Traps. Magic has a quite a few more. Since lands were already addressed, the others are, in no particular order:
Creature: Just like Hearthstone, these are your army. You use them to attack, and defend from attacks. They have power (combat strength) and toughness (hit points).
Enchantment: Spells that effect the battlefield as long as long as they're in play. Some have passive effects, some have interesting abilities. An enchantment that only affects a single creature is called an Aura.
Sorcery: Like Spells in Hearthstone. They can only be played on your turn during your main phases, have an immediate effect, and are placed in the graveyard afterwards.
Instant: Like sorceries, they have an immediate effect and go to your graveyard afterwards. However unlike sorceries, they can be played at any time, even on your opponent's turn. So long as you left lands untapped for them.
Artifact: Like enchantments, they stay on the field and have an effect, or have an ability. Generally, an artifact is a "thing" where an enchantment is "magic." Artifacts are usually colorless. Creatures can also be Artifacts, and Equipment artifacts can be given to a creature like weapons can be given to heroes in Hearthstone.
Planeswalker: A special "ally", usually famous characters in MTG lore. Each has loyalty, and can use one ability once per turn, which changes their loyalty. Planeswalkers must have at least as much loyalty as the ability uses if it removes loyalty, and a planeswalker with 0 loyalty is dead. They cannot usually attack or block like regular creatures.
