The words wait, weight sound the same but have different meanings and spellings. Why do wait, weight sound the same even though they are completely different words?
The answer is simple: wait, weight are homophones of the English language.
To remain or rest in expectation: waiting for the guests to arrive. See Synonyms at stay1.
To tarry until another catches up.
To remain or be in readiness: lunch waiting on the table.
To remain temporarily neglected, unattended to, or postponed: The trip will have to wait.
A measure of the heaviness of an object.
The force with which a body is attracted to Earth or another celestial body, equal to the product of the object's mass and the acceleration of gravity.
A unit measure of gravitational force: a table of weights and measures.
A system of such measures: avoirdupois weight; troy weight.
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").