What is the difference between the words "as" and "like"

What is the difference between as and like?
What is the difference between such as and like?

I take many private lessons like (such as) English, French and chemistry.
He runs like a horse.
He runs as a horse.

Are these sentences correct and why?

According to the OED definition of "like":

A.1.c In modern use (with following dative) often = ‘such as’, introducing a particular example of a class respecting which something is predicated.

   1886 Stevenson Lett. (1899) II. 41 A critic like you is one who fights the good fight, contending with stupidity.    1887 Colvin Keats i. 1 A birth like that of Keats presents to the ordinary mind a striking instance of nature's inscrutability.

So there is essentially no difference between "like" and "such as" when used as you described. The two have been interchangeable since the late 1800s.

Like and as both have many uses. Rather than try to list them all, I’ll focus on the ones that overlap, or nearly overlap.

• Both are used in comparative expressions. Like-phrases just indicate similarity: She moves like a cat. As-phrases usually refer to gradable adjectives, and they have a more specific meaning: The fence is as tall as a giraffe means that the fence’s height is ≥ the giraffe’s height.

• Both are used in similes (crazy like a fox, tight as a drum) which are after all just explicit comparisons. For the most part, these follow the rules above (if I am crazy like a fox, then I am crazy in the same manner as a fox, not at least as crazy as a fox).

• Such as and like are both used in noun phrases, postpositively, to add one or more examples (a distinguished mathematician such as Gauss or Möbius). In that meaning they are synonymous, the only difference being that such as sounds more formal. However, like can be used similarly outside of noun phrases; such as can’t.

Like Dale Chihuly, he employs a small army of artists. (correct)

*Such as Dale Chihuly, he employs a small army of artists. (wrong)

There are also verbs that combine specially with like and as (look like, pose as, etc.) These are just like any number of verb-complement combinations in English (think of, help with, etc.) You just have to learn them.

The NOAD reports the following note about the usage of like.

The use of like as a conjunction meaning as or as if ("I don't have a wealthy set of in-laws like you do;" "they sit up like they're begging for food") is considered by many to be incorrect. Although like has been used as a conjunction in this way since the 15th century by many respected writers, it is still frowned upon and considered unacceptable in formal English. In more precise use, like is a preposition, used before nouns and pronouns: "to fly like a bird;" a town like ours."

Informally, like is used as preposition with the meaning of "in the same way that," "as".

I felt like I'd been kicked by a camel.