Both to feign and to feint seem to mean to pretend. Are they synonyms or the same word with only a different spelling?
Or is there an actual concrete difference in their meaning?
A feint (noun) is primarily a deceptive move, such as in fencing or military maneuvering. It can also mean presenting a feigned appearance. Feint can also be a verb, but in that case it simply means to execute a feint.
To feign (verb) is to deceive; either by acting as if you're something or someone you're not, or lying.
There is some overlap between particular meanings of the two words (For example, his ignorance was a feint, he was feigning ignorance), but mostly they are separate.
Both words come from the French feindre, which means to "pretend, represent, imitate, shirk".
A feint is the result of feigning, just as a gift is the result of giving. Feigning an attack (originally in fencing, but extended to other military contexts, and then generally) took on a life of its own, so that a single word was needed; feint as a verb seems (seemed?) the obvious choice.