I wonder what the difference is between cleaning and cleansing solutions. Can anyone help?
I would say that "cleaning" normally involves removing dirt and blemishes from the surfaces of an object, while "cleansing" refers to removing impurities from the object as a whole.
To borrow one of @Damkerng's examples, an herbal remedy may claim to "cleanse the body, removing harmful toxins"; I would take that to mean that if I ingest it, it will eliminate said toxins from my innards. If the herbal remedy claimed only to "clean your body, removing harmful toxins", I would interpret that as something to put in my bath so that my skin no longer has harmful toxins clinging to it; a much less powerful statement.
Similarly, a product that cleans my skin may leave it with no dirt but all the pimples, warts, corns, age-spots, and wrinkles that were always there. A product that cleanses my skin should at the very least be eliminating all of the conditions that would normally cause pimples, warts, etc. to start forming (blocked glands, build-up of dead skin cells, etc.), if not actually eliminating the blemishes themselves.
Both clean and cleanse have a similar meaning: to make something or someone clean. Even Merriam-Webster Dictionary gives the definition of cleanse as "clean".
Personally, when someone uses the word cleanse, it usually means something beyond just clean. I usually take it as to make it pure or to make it clear. Here are things that I usually found the word cleanse is more preferred: cleanse your face (as in cleansing lotions or cleansing solutions in the OP's question), cleanse your soul, and data cleansing.
Merriam-Webster's Learner's Dictionary gives more details on the usage of cleanse.
usage The verbs clean and cleanse both mean “to make (something or someone) clean.” Cleanse usually refers to making the body or part of the body clean.
▪ cleansing the skin ▪ The herb is believed to cleanse the body of toxins. It can also refer to making a person's mind, soul, reputation, etc., clean. ▪ The ceremony is meant to cleanse people of their guilt and sin. ▪ Try to cleanse your mind through meditation. Clean is more common than cleanse and its use is less specific.