How does a patent differ from a copyright?

Copyright protects a creative work, fixed in a tangible medium. Patents protect an invention, including a method, article of manufacture, machine, drug, etc.

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Copyright is automatic, as soon as you create the work. It can also be registered by providing a copy of your work to the Library of Congress (in the US) or similar entity in other countries, which provides you with the ability to sue for statutory damages for infringement, collect attorney fees, etc. Copyright is cheap, around $35 to register a work, and lasts for the lifetime of the author plus 70 years. BUT, it doesn't stop someone else from making the same exact work independently - only if they actually COPY you. For example, if someone lived in a cave for years and independently, without ever hearing about Harry Potter, wrote a story about a boy wizard with a scar who visits Bogwarts Castle and fights the unmentionable one, that wouldn't infringe Rowling's copyright (but good luck proving that they never heard of it).

Copyright is very useful where your specific work is desired, such as a AAA game, a famous work of art or novel, a tv show or movie, music from a specific artist, etc.  It isn't useful where consumers want the functionality, like "a mobile match-3 game" or "a word processor" but don't care who it comes from. For example, copyright is great for protecting "Star Wars: The Force Awakens" because consumers want that rather than the knock off "Space Game: Power Wakes Up"... but it isn't that helpful for protecting "Tiny Tower" vs. "Dream Heights" vs. "Sim Tower" vs. "Tower Game", etc., etc.

Patents, on the other hand, protect implementations of the underlying idea. A patent prevents others from making, using, selling, or importing the invention, even if they independently create it.  Patents prevent those knock offs - it doesn't matter what you call your machine; if it does the same thing, in the same way, with the same parts, as someone's patented machine, it infringes. Patents are much more powerful than copyright in that way... but they only last 20 years from the date of filing, and can cost between $10-25k to obtain, sometimes more, and it may be several years from filing before they are issued. Patents are examined by patent examiners with experience in the relevant industry, and are only issued when the examiner is assured that the invention is new, non-obvious, and sufficiently described in the patent application to enable another person of skill in the art to make and use the invention.

Patents and copyrights can cover the same thing - software is copyrighted as soon as it's written, since it's a work of creativity; but it can also be covered by a patent on the functionality. This overlapping protection is useful for protecting against different types of infringement, from piracy to competitors.

Related post: What is difference between copyright and patent?