The fact is that I use semicolons a lot more than most contemporary writers, mostly for what I consider good reasons.
If you read books printed in the 19th century (as opposed to modern editions with normalized punctuation) you'll find that semicolons were formerly used quite often where we would now use commas.
Today, many people do not use semicolons at all—some because they are confused about how to use them and some because they rarely write the sort of complex sentences that requires them.
In fact, I would guess that the main use of semicolons in nonprofessional prose is as a winking emoticon [;]. And I'd bet that more people wink on the Web using semicolons than wink with an eye in real life.
But back to semicolons and “but.”
One argument is that “but” often serves the same purpose as a semicolon, and that in those cases you can’t use them together. I agree.
The excellent Purdue English OWL site gives these examples of semicolon use:
It rained heavily during the afternoon; we managed to have our picnic anyway.
It rained heavily during the afternoon; however, we managed to have our picnic anyway.
This could also be rendered “It rained heavily during the afternoon, but we managed to have our picnic anyway.”
Many people will go further to say that when two independent clauses are linked by a coordinating conjunction like “and” or “but” a semicolon should never be used, but they often fail to mention the exceptions.
The Chicago Manual of Style explains this in entry 5.70:
If the clauses of a compound sentence are very long or are themselves subdivided by commas, a semicolon may be used between them even if they are joined by a conjunction: “Margaret, who had already decided that she would ask the question at the first opportunity, tried to catch the director’s attention as he passed through the anteroom; but the noisy group of people accompanying the director prevented him from noticing her.”
This is actually more complex and subtle than it looks at first. Note that the CMS says the semicolon may be used, not must be used. This is not a matter of absolute correctness, but of style preference.
Also note that the example given combines two points: an initial clause which is either long or subdivided with commas. Some people think you should apply this rule only when the initial clause is so long and complex as to make the sentence confusing without the semicolon. I lean toward the view that whenever the conjoined clauses are subdivided by commas, you need to escalate to a semicolon.
An example from my Web site:
If you're selling something, it's for sale; but if you lower the price, it goes on sale.
In this sort of sentence, common on my site, a compound sentence consists of two balanced halves each of which has a comma subdividing it. It seems to me logical to escalate to a semicolon between the clauses to distinguish that break from the smaller breaks to either side of it.
Another example from the entry in “input”:
Be aware that it’s not welcome in all settings; but whatever you do, don’t misspell it "imput."
Here only the second clause is subdivided by a comma. Many people would not use the semicolon at the conjunction because it’s not needed for clarity’s sake, but I prefer to be consistent.
In the end, this is a matter of preference. If you’re being edited or graded, you need to know what the appropriate pattern is; but when you’re writing to suit yourself you can prefix a semicolon to a “but” when it feels necessary.
All that said, I did find a number of cases on my site where I had gone overboard, so I changed quite a few semicolons to commas before “but.”