One difference is that “thorough” is most often used to modify verbs, while “complete” can be used to modify both verbs and nouns. Example: “He gave the kitchen a thorough/complete cleaning.” Here, either is OK. But you (usually) would not use “thorough” to modify a noun. Example: “My son has a complete set of Pokémon cards.” You would not use “thorough” for this sentence. So, maybe it is useful to you to think of “thorough” as describing an action that is well and completely done. “Complete” can describe either an action that is fully done, or a physical thing or set that is not missing any part. Another example: “The archeologists found a complete skeleton of a dinosaur.” Here, only complete works, not thorough. But you could say “He gave the police a complete/thorough description of the crime.”

"Complete" is a more generic term that describes that you have finished all components of a task. "Thorough" means that you went to painstaking detail through all components of a task. Example sentence: The sloppy janitor completed cleaning the bathroom but he did not do a thorough job. There was some toilet paper left on the ground.