Not only are whine and whinge spelt similarly, they also have incredibly similar meanings. Both words have noun and verb forms, which makes them especially difficult to differentiate Both words mean to complain, but it is the manner of complaint which separates the two.
According to Wordnik, whinge means "to complain, especially in an annoying or persistent manner." Merriam-Webster defines whinge as "to complain fretfully." This is to complain in a manner in which one is emotionally strained.
Whine is defined in Merriam-Webster as "to utter a high-pitched plaintive or distressed cry," or "to complain with or as if with a whine." This means that "whine" can be a distressed cry, or to complain in a distressed manner. For instance, a dog can let out a whine, which lets its owners know that it is in distress.
Whinge is British
According to Merriam-Webster, whinge is a British word. Which means, although it is a proper English word, it is mostly used by British English speakers.
Whine as defined by WordReference.com
WordReference.com adds an additional definition of whine that separates it from whinge. It defines whine as "a feeble or petulant complaint." A feeble or petulant complaint is a childish, weak, bad-tempered complaint.