The spell-check application of most word processing software programs would not catch a slip-up of these two words. Spell-check is looking for words that aren’t in its dictionary, and words that resemble words in its dictionary but are possibly spelled wrong. Spell-check isn’t perfect. It doesn’t know and can’t guess what word you wanted or what word you meant, it can only judge the words on the page. If you used words that are all spelled correctly, it gives you a pass anyway.
Autocorrect suggests words that start with the same letters. It suggests words to try to save you time. Sometimes it’s right, but frequently it’s wrong and very funny.
Disparate is an adjective. It means two things or ideas that are very different.
Desperate is an adjective. It means a person feeling lots of hopelessness and frustration. Desperate people perceive limited choices and options.
The following story uses both words correctly:
Desmond and Destiny were identical twins, but they had very disparate personalties. Desmond, a star athlete, seemed to enjoy school and its social scene than his sister, a gifted music student. Destiny was desperate to receive as much attention as her brother did. It finally happened when she was accepted to Juilliard.