The words discreet, discrete sound the same but have different meanings and spellings. Why do discreet, discrete sound the same even though they are completely different words?
The answer is simple: discreet, discrete are homophones of the English language.
Marked by, exercising, or showing prudence and wise self-restraint in speech and behavior; circumspect.
Free from ostentation or pretension; modest.
Constituting a separate thing. See Synonyms at distinct.
Consisting of unconnected distinct parts.
Mathematics Defined for a finite or countable set of values; not continuous.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").