To begin with, let’s shake the dust out of the curtains fast and furious: it is right and proper and historically justifiable to begin a sentence with but. And to begin a sentence with and. But that does not mean one should act with abandon in doing either. Both words are conjunctions, and what will decide where to place and how to punctuate them involves some sensitivity to the relationship between the ideas they are connecting.
Here is a sentence from Oscar Wilde’s sweet little story The Happy Prince. The prince of the tale is a statue that stands high over a city, and a swallow has taken nest at his iron feet: immobility meets freedom. After a time of the bird’s coming and going, the Prince says to the swallow, “you tell me of marvelous things, but more marvelous than anything is the suffering of men and of women.” There is a fluidity to the sentence; it runs easily between the idea of marvelous things and the idea of even more marvelous things; and it flows with some energy because these two ideas are set gently against each other with the conjunction but preceded only by a comma, as it most regularly is. When conjunctions join clauses, each has a logical role to play. The conjunction but (along with however, nevertheless, still, and some others) is meant to put ideas into contrast or opposition.
What we should see, though, is the manner of the opposition. The Prince, it seems, is instructing the swallow a little by intensifying the idea of marvelousness: the first clause of the sentence speaks of marvelous things and the second clause speaks of things more marvelous; the same adjective appears in both clauses, and the only difference between them is their degree. The contrast which the conjunction but is meant to signify is not great or dramatic; it is instead, we might say, organic: it builds more out of the same thing, the ideas of both clauses, the marvelous, are both of a piece. Marvelous things, more marvelous things: they are all, though, marvelous. We feel this gentle increase in part because the two clauses are never severed from each other. They remain close to each other, staying within the bounds of the sentence and joined in a moderate contrast only by a comma and the conjunction.
Compare this now to a scene earlier in the story. The swallow is complaining to the Prince that some young boys of the city were throwing stones at him: “They never hit me, of course; we swallows fly far too well for that, and besides, I come of a family famous for its agility; but still, it was a mark of disrespect.” What is the conjunction but now putting into contrast and why is it preceded by a semicolon, not a comma? The opposition here is more involved than the simple change of degree which we saw in the first example. The disrespect that is named in the final clause is being set in contrast to a complex of ideas: never being hit, flying too well, famed lineage. Far from being an organic outgrowth, these three ideas stand up against something quite different from them: disrespect. The contrast is sharp, rough edged, and in order to convey that severity, the final clause has been isolated by means of a semicolon. We are to see forces arrayed against each other, and the fight is not a fair one: unharmed skill and illustrious ancestry are together too much for callow youth.
And right here we can feel what effect is achieved in beginning a sentence with but. Had Wilde written two sentences for the one he did, the isolated final clause would have been disassociated entirely: They never hit me, of course; we swallows fly far too well for that, and besides, I come of a family famous for its agility. But still, it was a mark of disrespect. An entirely acceptable arrangement, but one that would have heightened the contrast to a degree the author could not have wanted. Beginning the second sentence with the conjunction throws more than a pause between the contrast: it opens a gap, a breach into which the reader can hear a faint tone of self-doubt, of concession, of yielding—ideas far in truth from the proud, self-knowing swallow.
Conjunctions, then, may begin a sentence, but we should be aware of the strength they carry in doing so. Writers (and all artists) work by making choices, and we as readers are to read their choices aright to reap their ideas in full.