を元(もと)にBased on, From, According to

These phrases both mean 'based on' and can overlap, but differ slightly. にもとづいて is more formal and tends to show a concrete or logical foundation for an action or product. をもとに is less formal (but still used in formal contexts) and can indicate a general basis or inspiration where にもとづいて may not fit.

Both involve a basis or source, but differ in usage. によると・によれば reports information, e.g. 天気予報によると雨が降るそうです ('According to the weather forecast, it will rain.') をもとに means doing something based on a source or foundation, e.g. 事実をもとに小説のストーリーを考えた ('I thought up a story for a novel based on true events.')

をもとに and にしたがって can both be translated as according to or based on, but differ in nuance. をもとに suggests a foundation or inspiration, as in 実話をもとに書いた小説 (a novel written based on actual events). にしたがって focuses more on being in line with or following something, as in 指示にしたがって (according to instructions). It can also mark proportional changes, as in 普及にしたがって (as it spread).

をもとに indicates a fixed source or basis for something, as in 実話をもとに小説を書く (Write a novel based on true events). に応じて shows flexible adjustment or response to conditions, as in 状況に応じて対応を決める (Decide the response according to the situation).