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I think they are not. They can’t be exactly interchangeable, otherwise one of them would sooner or later die out.
In Oxford Advanced Learner’s Online Dictionary the verb “cleanse” has 2 meanings, whereas the verb “clean” has 4. It entails that clean is more than cleanse, it means it’s used in a wide variety of contexts. The primary meaning of clean is “to make something free from dirt or dust by washing or rubbing it”, whereas that of cleanse is “to clean your skin or wound”. So, it can relatively be stated that cleanse is ‘narrower’ than clean.
As for its being formal, at first blush, cleanse does sound more formal. However, I think the difference lies in meaning, not in register.
They literally mean the same thing but there are connotations that differ. Clean means something more immediate and superficial. Cleanse means a deeper more thorough cleaning, often a complete cleaning; e.g. I will clean the garage but I am cleansing the house of every speck of dirt.
Another difference is that clean is almost always used literally but cleanse is often used figuratively as in cleansing your soul or ethnic cleansing.
These items I have mentioned are not hard and fast rules. You will see them occasionally used exactly opposite from my description, but I believe these are the more common ways of using these words.