謝邀。
如果您要較簡單單一, 非此即彼,"正確"跟"錯誤"二分的答案, 那麼 ---
至晚在十九世紀, 而且直到廿世紀, 幾個有影響力的英文 規範語法派/prescriptivists, 他們所寫的, 發行量頗大的冊子上, 規定不可以, 規定那是錯誤。仍頗有些人受他們影響, 認為不可以, 因而看到有人用時, 會當成"錯誤"而皺眉頭; 而受到影響的人當中, 有可能會有您的老師, 因而或許會有明顯直接後果 -- 扣分。所以, 尤其使用正式英文時, 特別是一開始使用英文, 在句首不要用but 跟 and。
BUT,
那個"規定", 跟英語母語人士即便在使用在發佈前 可以好整以暇自行目測檢視的英文時, 所呈現
的實際現象不符合, 如同你已有注意到的現象; 遑論是以直覺用口語立即直接反應來使用的英語,
所呈現的語言現象。而那個"規定"也不是所有語法學者均認可是鐵律。見過甚至有認為
那是莫名其妙的"規定"者, 尤其較傾向語言描述派/descriptivist的學者更有認為如此。
想要對這個現象多了解, 可Google: "starting a sentence with but", 便可獲得更多資料, 會讓
您在使用英文英語時, 好像會變更難更複雜的資料(接受並執行黑白分的單一規定, 會似乎較簡單),
但卻可以讓您在這一點的認知上, 跟實際語文現象, 跟英語文這個語言的實際現象符合的資料。
这时候请参考投稿处的style guide，法无禁止即允
个人拙见，浅薄涉猎，毫无深度，仅供参考。
1. and 和 but 和可以放在句首用，即使在学术文章之内（注意黑体）
A conjunction connects sentences, clauses, or words within a clause{My daughter graduated from college in December. And my son will graduate from high school in May.}(and connects two sentences);{I said hello, but no one answered}(but connects two clauses);{we’re making progress slowly but surely}(but joins two adverbs within an adverbial clause). Conjunctions connect pronouns in the same case{he and she are colleagues}{the teacher encouraged her and me}.
摘选自 CMOS 5.192 Conjunctions defined
There is a widespread belief—one with no historical or grammatical foundation—that it is an error to begin a sentence with a conjunction such as and, but, orso. In fact, a substantial percentage (often as many as 10 percent) of the sentences in first-rate writing begin with conjunctions. It has been so for centuries, and even the most conservative grammarians have followed this practice. Charles Allen Lloyd’s 1938 words fairly sum up the situation as it stands even today:
Next to the groundless notion that it is incorrect to end an English sentence with a preposition, perhaps the most wide-spread of the many false beliefs about the use of our language is the equally groundless notion that it is incorrect to begin one with “but” or “and.” As in the case of the superstition about the prepositional ending, no textbook supports it, but apparently about half of our teachers of English go out of their way to handicap their pupils by inculcating it. One cannot help wondering whether those who teach such a monstrous doctrine ever read any English themselves.[1]Still, but as an adversative conjunction can occasionally be unclear at the beginning of a sentence. Evaluate the contrasting force of the but in question, and see whether the needed word is really and; if and can be substituted, then but is almost certainly the wrong word. Consider this example: He went to school this morning. But he left his lunch box on the kitchen table. Between those sentences is an elliptical idea, since the two actions are in no way contradictory. What is implied is something like this: He went to school, intending to have lunch there, but he left his lunch behind. Because and would have made sense in the passage as originally stated, but is not the right word—the idea for the contrastive but should be explicit. To sum up, then, but is a perfectly proper word to open a sentence, but only if the idea it introduces truly contrasts with what precedes. For that matter, but is often an effective word for introducing a paragraph that develops an idea contrary to the one preceding it.
[1]Charles Allen Lloyd, We Who Speak English: And Our Ignorance of Our Mother Tongue (New York: Thomas Y. Crowell, 1938), 19.
摘选自 CMOS 5.206 Beginning a sentence with a conjunction
简评：CMOS 定义 conjunction 可以连接两个句子而不仅仅是分句，所以在大部分符合 Chicago Style 的学术文章中，这不算是错误的。且虽然 MLA Handbook，The Publication Manual of the APA，ACS Style Guide，AMA Manual of Style 等这些别的 Style 的规定我没有看过，但是在很多 MLA Style 和 APA Style 和别的一些 Style 的学术文章中也多见 But 可以放在句首的情况。CMOS 是所有 Style Guide 里面对于语法规定最为详细的一本，所以我觉得如果不细究，那么 CMOS 中的定义是可以通用的。其次，CMOS 在 5.206 对于 but 的使用做了一些意义上的规定，我已经用黑体标出。可以看出 CMOS 的态度是支持连词用在句首并且引用了段落嘲讽反对连词放在段首的人。但是，不是所有情况下的连词尤其是 But 放在句首都是合理的（虽然语法上不算错）。
Kai Wang 举得例子里那些短句子，自然不适合独立成句然后用连词连接，用副词还会更怪，所以只会给人一种连词放句首会不好的错觉。
就说 We started to go home. But we ran out of gas. 这一句。改成 We started to go home, however/nevertheless, we ran out of gas. 根本就不如 We started to go home, but we ran out of gas. 要简洁，要好。两个分句虽然都可独立成句，但是意思却可以直接连成一个句子，直接用 We did't go home because we ran out of gas. 也是可以的。所以这个例子很牵强。只能说是 But 不适合放在句首的一个例子，而不能代表整体情况。
一般性的学术范围内以 But 开头的句子不会是这种句子，而多会是段首转折上文所讲的内容或者因为前一句过长，不宜作为单独的分句和 but 一句连成一句话。
我举一段文字的例子，文意还颇具讽刺性：
[But ordinary concepts are notoriously vague; individual conceptions and linguistic
usage varies widely.]Moreover, inquiry often demonstrates that the ordinary
concepts used initially to frame a project are not, as they stand, well-suited
to the theoretical task at hand. (This is one reason why we may shift from a
conceptual project to an analytical one.){But precisely because our ordinary concepts
are vague} (or it is vague which concept we are expressing by our everyday
use of terms), there is room to stretch, shrink, or refigure what exactly we are
talking about in new and sometimes unexpected directions.
Haslanger, Sally. "Gender and race:(what) are they?(What) do we want them to be?." Nous 34, no. 1 (2000): 31-55.
[……]部分就是转折文章将 But 放在段首的例子
{……}部分就是用来和前文过长的分句断开的例子，换成是关系副词一下语气就降下来了，相比之下，But 在这里作为句首的是完全合理的
为了避免有人对于该段落的代表性做出质疑，我对本段落作者做个简单介绍：
Sally Haslanger is a professor in the Department of Linguistics and Philosophy at the Massachusetts Institute of Technology.
所以该段落应该还是有一定的代表性的，因为作为一个现代语言学家写的文章，肯定在语法和形式上都要比一般人严谨且应该更符合现代英语语言的整体趋势。
所以我初步的观点是，在学术或正式文章之中，连词可以放在句首或者段首。只要意义上合适即可，不必刻意将本来单独作为分句更合适的句子独立成句，也不必逢段首或者句首就必须将本来想用的连词替换做关系副词。在使用连词或者关系副词之前进行想一下哪种情况更好，如果将转折部分独立成句不如作为分句使用，那么当然要用两个分句的形式。这时候就不涉及对错的问题了，而是好坏程度上的问题；合理的运用连词和关系副词是对的，不需要用「连词不适合放在句首或者段首」的规定来框住自己。只要放在句中合适，且相比使用分句或者使用关系副词，能更加简洁清晰地表达你所表达的内容，为什么不用？
2. 具体环境具体参考
普通文章中
Starting a sentence with a conjunctionOxford Dictionary Conjunctions
You might have been taught that it’s not good English to start a sentence with a conjunction such as and or but. It’s not grammatically incorrect to do so, however, and many respected writers use conjunctions at the start of a sentence to create a dramatic or forceful effect. For example:Beginning a sentence with a conjunction can also be a useful way of conveying surprise:
- What are the government’s chances of winning in court? And what are the consequences?
And are you really going?
- But didn’t she tell you?
- It’s best not to overdo it, but there is no reason for completely avoiding the use of conjunctions at the start of sentences.
普通文章中或者日常书写的情况下，当然要求就更少了，不必要纠结于语法细则。当然，大部分权威语法指导书籍都没有将连词放句首作为语法上的错误现象。
另外根据题主的描述的两种情况来看：
The New York Times Manual of Style 没有对于 but 在句首使用的合理性作出规定。
但是：
But 作为句首连词使用在 The New York Times Manual of Style 几乎每两页就会出现一次，比如：
所以基本上可以说 The New York Times 是默许 But 作为句首或者段首词出现的，当然，当使用分句更合适的情况也不会刻意要求去让转折部分独立成句。
同样 The Economist Style Guide 之中也没有对于 But 是否可以作为句首或者段首词出现的规定。
但是：
以下是 The Economist Style Guide 中的句子（仅取 A - B 的句子），格式没有做任何修改：
Some are short and to the point (so prefer lay off to make redundant). But many are unnecessarily long (so use and not additionally, car not automobile, company not corporation, court not courtroom or courthouse, transport not transportation, district not neighbourhood, oblige not obligate, rocket not skyrocket, stocks not inventories unless there is the risk of confusion with stocks and shares).
An should be used before a word beginning with a vowel sound (an egg, an umbrella, an MP) or an h if, and only if, the h is silent (an honorary degree). But a European, a university, a U-turn, a hospital, a hotel. Historical is an exception: it is preceded by an, the h remaining silent.
A Paris-based group may be all right, if, say, that group operates abroad (otherwise just say a group in Paris). But avoid community-based, faith-based, knowledge-based, etc.
If Belarusians (not Belarussians) wish to be known by the Belarusian form of their names (Ihor, Vital, and Life-President Alyaksandr Lukashenka), so be it. But use the familiar, Russian placenames (Minsk not Miensk, Brest not Bryast, Gomel not Homel).
AP Style 中也没有对于 But 的使用作出任何规定。
从上面的例子中可以推断，连词可用在句首被默认是语法正确的一种合理的形式。然而运用过程中，大多数情况都是段首转义或者是因为前一个分句过于冗长；少数情况下是为了加强语气。
对于另外的英文媒体来说，英国很多媒体的 Style Guide 和美国有不同的规定。
美国直接参考 The New York Times Manual of Style 和 AP Style 就大致可以了，英国参考 BBC，Guardian and Observer 还有 The Economist 各自的 Style Guide，卫报的比较全面一些。
反正我粗略一看都是没有关于 But 是否可以用在句首的规定。
结论：
连词可以放在句首使用，包括 and，but，so 等等，语法上来讲都不算错误。所以不必要用「but 不适合放在句首」这种规矩框住自己的写作。但是如果所写的句子作为两个分句来说更加合适，则适合直接用两个分句；其他情况下，当使用 but 作为一个句子的开头不会使得表意不明，不必硬要将连词换成关系副词。其次还需要参考你发稿的媒体的相关规定，或者论文发表的一些相关标准。就我目前看到的情况，还没有任何规定说正式文章使用 But 作为句首是一种不合适的行为，说这种形式不适合出现在学术文章或者正式文章中不过都是个人的偏好罢了。当然，如果你有一个刻板的导师或者说审阅你文章的人，那么最好还是随着他的套路走。现在还有很多人觉得 get 表达获得这个意义时还必须改为 gain，所有第一人称跟随 w 开头的助动词必须换成 sh 开头，表达要求允许必须用 may/might 而不能用 can/could。遇到这种人，就不要拿出各类 Style Manual 来驳他了，随他来吧。
我认为可以。
有些人认为，出现and或but在句子开头的情况时，要么把and或but前面的逗号改成句号，使得它与前面的句子连成一句话；要么就把and或but改成moreover或however之类的副词。
但是有这么一种情况，就是and或but前的句子已经很长了，递进或转折后的部分有必要另起一句（比如本句以及下一句）。而过多地使用moreover或however之类的副词，会让人感觉语气很奇怪、很墨迹，尤其会有一种故作客观的感觉。相信也一定有些人会给出“不要过度使用moreover、however等副词”这样的写作风格建议。
我觉得，在比较正式的文体中，遇到在句子开头需要使用and或but的情况时，可以把适量的and或but换成moreover或however之类的副词；在非正式的文体中，但用无妨。
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在网上看了一些讨论之后，我发现有些人提出“不能在句子开头使用and和but”的原因是，有些人会把本该放在一句中的两层意思（都比较简短）拆成两句，使得行文显得不连贯。对于英文写作尚不熟练的人来说，这的确是应该注意避免的。但是，把它推广成为铁律，则矫枉过正了。
下面有翻译:)
When I was in elementary school, I was taught this rule too. I think it was taught this way so we would learn the function of conjunctions - "joining words".
There is no grammatical problem with starting a sentence with "but" or "and". But I would suggest avoiding overusing it in academic writing. You can use "however" or "additionally" instead.
And I hope this is helpful. :)
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栗子的知乎答案索引：栗子树
栗子的微信公众号：E-Speller 或者 栗子英文 （只有精彩的原创）
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谢谢 @乙也 的翻译：
这个规则是我上小学时候学的，我认为这样的目的是让我们学习连词的作用。（Joining Words，连接词，包括连词与连接性副词）。
用“但”或“和”开头没有语法问题，但我建议避免在学术写作中过度使用，可以使用“However”或“Additionally”来替换。
希望能有所帮助。 :)
