The sentences “He speaks English like an Englishman” and “He speaks English as an Englishman” mean two different things altogether.
Depending on context, the conjunction “like” can mean “as if” or “in the same way that” or “in a way or manner that.” In the first sentence you presented, “He speaks English like an Englishman,” “like” is used in the sense of “as if,” so the sentence is synonymous with “He speaks English as if he is an Englishman.” Aside from this sense, “like” can also be used in the sense of “in the same way that,” as in the sentence “They detest each other like children do bad-tasting medicine”; and also in the sense of “in the way or manner that,” as in the sentences “He negotiates deals like an old entertainment impresario should” and “She testified unconvincingly in court like you told me.”
In contrast, in the sentence “He speaks English as an Englishman,” the conjunction “as” is used in the sense of “in the way or manner expected of”—meaning in this particular case that the speaker is, in fact, an Englishman and speaks English in the way or manner expected of an Englishman. This sense is entirely different from that of the sentence “He speaks English like an Englishman,” which means that the speaker is, in fact, not an Englishman but can speak it as well as an Englishman does.
