The difference between a logic board and a motherboard is subtle but simple. Motherboard is a generic term, while a logic board is a brand-associated term.
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Motherboards are also known as logic boards, system boards or main boards. A common abbreviation for it is "mobo." Motherboards can be found in a number of different types of electronic devices, from televisions to computers, and they are the main component to which other parts are connected.
The term "Logic board" was coined by Apple in the early 1980s for the motherboards in their Macintosh computers. The phrase stuck, and they are still called logic boards to this day despite being basically the same thing as a motherboard.
A "Motherboard" is a more generic term for the same thing as a logic board. The only notable difference is a logic board is generally considered to be Macintosh, whereas a motherboard could be a Mac, PC or any other computer. The same components plug into both, like CPU, RAM, graphics cards, hard drives and optical drives.