Norman Schur, British English A to Zed, third edition (2007) reports that fillet in British English has two corresponding senses in U.S. English: tenderloin and filet. Here is Schur's entry for fillet:
fillet, n. 1. tenderloin 2. filet
(Rhymes with MILLET, not MILLAY.) On an American restaurant menu the equivalent would be tenderloin steak, or perhaps filet mignon. The term may also be applied to pork, lamb, etc. 2. a piece of fish served without the bones.
Merriam-Webster's Eleventh Collegiate Dictionary (2003) treats filet as a variant spelling of one particular sense of fillet. Here is the Eleventh Collegiate's entry for fillet:
fillet in sense 2b also filet n {ME filet, fr. AF, dim. of fil thread, fr. L filum — more at FILE} (14c) 1 : a ribbon or narrow strip of material used esp. as a headband 2 a : a thin narrow strip of material b : a piece or slice of boneless meat or fish; esp : the tenderloin of beef 3 a : a concave junction formed where two surfaces meet (as at an angle) b : a strip that gives a rounded appearance to such a junction; also : a strip to reinforce the corner where two surfaces meet 4 : a narrow flat architectural member: a : a flat molding separating others b : the space between two flutings in a shaft
So in Schur's view, the main difference between fillet and filet is that, in addition to being coextensive with filet in a general sense in connection with cuts of meat, fillet in British English is used to refer to the specific cut of beef called, in U.S. English, tenderloin.
For its part, Merriam-Webster emphasizes that fillet and filet have the same meaning in meat sense, but then notes that fillet has multiple additional meanings that filet does not. On the other hand, according to the Eleventh Collegiate, filet has one meaning in English that fillet doesn't:
filet n {F, lit., net} (1838) : a lace with a square mesh and geometric designs
Aside from an entry for filet mignon (which Merriam-Webster defines as "a thick slice of beef cut from the narrow end of a beef tenderloin" and assigns a first occurrence date in English of 1835), that is the only entry for filet in the Eleventh Collegiate.
With regard to the word's etymology, Glynnis Chantrell, The Oxford Dictionary of Word Histories (2002) has this entry for fillet, confirming the summary that appears in the Eleventh Collegiate:
fillet {Middle English} Early use of the word was to denote a band worn round the head. Old French filet 'thread' based on Latin filum 'thread' is the source of the word in English. Current senses all involve the notion 'thin strip' (e.g., fillets of beef, fillet impressed on a book cover, etc.).