In "Dover Beach," the tone is quite serious. The speaker compares the image of the sea as a calming presence. Arnold refers to the "Sea of Faith." Just as the oceans connect all the continents and all the people, religion once connected all people in the same spirit of something greater than themselves. Since the speaker (and Arnold himself) no longer believes in the power of religion (Christianity) to do this, he looks to his companion (his lover) for some meaningful connection.
In "The Dover Bitch," the speaker is flippant and irreverent. The poem reads as if the speaker is in a bar relating the story to a stranger in a very informal way. The speaker suggests that he knows Arnold (speaker of "Dover Beach") and the lover he speaks of. In fact, the speaker in "The Dover Bitch" even admits to having affairs with the girl (from "Dover Beach") himself.
In "The Dover Bitch," the girl is not thinking about the loss of religious belief and love as a substitute. She's thinking about French wine and perfume. This is pretty shallow when compared with the dramatic monologue of "Dover Beach." Or, you could say, it is pretty realistic. Maybe the girl is thinking about such things, not impressed with Arnold's pontifications about cosmic unity and love.
Compare these two passages, the first from "Dover Beach" is serious; the second, from "The Dover Bitch" is crude by comparison; but in mocking "Dover Beach," it can also be construed as humorous:
Ah, love, let us be true
To one another! for the world, which seems
To lie before us like a land of dreams,
So various, so beautiful, so new,
And he said to her, 'Try to be true to me,
And I'll do the same for you, for things are bad,
All over, etc., etc.'
The speaker of "Dover Beach" looks for romantic love to substitute for the lack of religious unity in the world. The speaker of "The Dover Bitch" is just looking for someone to spend time with (or for just the night). That's why he's dismissive and rude when speaking about the woman, but acknowledges that she is "dependable." The speaker in "The Dover Bitch" is crude by comparison, but, depending on your interpretation of the tone, he might also sound more realistic, at least in the informal way some people speak.