They do not have quite the same meaning. "Fallout" refers to the negative RESULTS of an event. For example, "The fallout from x was unexpected". "Aftermath" is the PERIOD following an unpleasant event. For example, "Many businesses went bankrupt in the aftermath of x".

The words have a very similar meaning - the negative result of an action. Often either word could be used. In general though, with aftermath, the action has now stopped, ie the aftermath of the war, flood, or earthquake. Fallout is generally about less severe actions, ie. the result of an incorrect statement made by a politician.

Thank you so much! Very helpful answers. Dictionary sometimes doesn’t give enough explanation to differentiate between synonyms.

I agree with Timothy that the difference in meaning is very small. That doesn't mean that you can use them in exactly the same situations though. For example, The psychological fallout of being obese (i.e. psychological difficulties BECAUSE OF being obese) You can't say "the psychological aftermath of being obese". I think this is because "aftermath" does contain a more specific sense of "after" in time, and a less direct sense of consequences. Another example: In the aftermath of the war, people continued to marry (despite their difficulties). That doesn't mean they got married because of the war. It just means we should bear the post-war context in mind when we consider these events.

Very little. I think of the word 'fallout' as more informal than 'aftermath,' and also as more negative, despite the definition at dictionary.com -- https://www.dictionary.com/browse/aftermath -- suggesting that 'aftermath' would be used most often for a bad result. Please note that the first definition of 'fallout' is unrelated. https://www.dictionary.com/browse/fallout