They are often used interchangeably.
But Alleviate is used more commonly in pain relief and in medical situations.
Mitigate is more commonly used in legal and conflict resolution situations.
Alleviate = to make something bad, less bad or painful, (etymologically from the word to make MILD why it is used in medical terminology)
Mitigate = To make something bad, less bad or painful,(etymologically from the word to RISE/make lighter why it is used in legal and conflict terminology)
For other meanings you can use either word, but try to remember "Alleviate is making milder" and "Mitigate" is making the situation lighter.
Would you make pain milder (alleviate) yes, or lighter (mitigate) no.
Would you make a conflict or problem milder (alleviate) no or lighter (mitigate) yes.
I hope this helps you when to choose alleviate or mitigate.
Always use "alleviate" for pain and suffering
Always use "mitigate" for legal and conflict resolutions.
For other situations remember the original words. I have told you to make milder or to make lighter in severity.