From eHow:
Comma (,)
1) Use a comma to separate 2 independent clauses joined by a coordinating conjunction such as and, but and or. The key here is to have 2 subject verb pairs.
Example: The students reviewed for the exam, and the teacher
corrected the term papers.
In this example we have 2 subject verb pairs: "students reviewed" and "teacher corrected."
Example: The teacher corrected the
papers and entered the grades in the
grade book.
In this example we use the coordinating conjunction "and" but have 1 subject and 2 verbs: "teacher corrected, entered"; therefore, no comma is used.
2) Use a comma to separate items in a series to avoid ambiguous meaning.
Example: Uncle willed me his property,
houses, and warehouses.
In this example, we mean that uncle willed me 3 items--his houses, warehouses, and property.
Example: Uncle willed me his property,
houses and warehouses.
In this example, we mean that uncle willed me 2 items--all his property, which consisted of houses and warehouses.
3) Use a comma with introductory elements such as subordinating clauses that come at the beginning of the sentence.
Example: Because the river had
flooded, the school closed for the
week.
The introductory element, or subordinating clause is "because the river had flooded." It is introduced with the subordinating conjunction "because."
Example: The school closed for the
week because the river had flooded.
In this example the subordinating clause is at the end of the sentence so we do not use a comma.
Semicolon (;)
1) Use a semicolon to separate 2 independent clauses in a sentence, closely related, with no coordinating conjunction.
Example: The students reviewed for the
exam; the teacher corrected the
papers.
Example: The candidates spoke to the
crowds during Election Day; each
candidate spoke passionately about the
fate of the country.
2) Use a semicolon to separate items in a series where the series themselves contain commas.
Example: We visited our relatives in
Albany, NY; Philadelphia, PA; and
Washington, D.C.
The comma is correct.
The semicolon would be correct if the first clause was a complete sentence, but it is not. It includes "not only," which calls for a linking word, such as "also." ("This not only produces higher success rates, it also increases....")
If you were to leave out "not only," the semicolon would become correct (but you would also lose the comparison you are trying to convey).
Best Answer
There are three and a half different ways to use however*. This one needs a semicolon.
The first is using it as a conjunctive adverb. In this sense the meaning of however is that the independent clause that follows counters the independent clause before it (denying it, giving a caveat, stating something as true that we would not expect considering the first clause, etc.)
As a fully independent clause, it needs a semicolon to separate them. (As per the link MετάEd gave in the comment on the question, though note that a conjunctive adverb is not the same as a conjunction, so the "Don't use it with with conjunctions" doesn't apply).
And with conjunctive adverbs of more than one syllable, we use a comma after it to help clarify what is the second independent clause.
That's use of however number 1. Number 1½ is that since they're independent clauses we can just have them as separate sentences:
We would favour a new sentence or a semicolon depending on whether we wanted the semicolon's suggestion of a particularly strong tie between the two clauses.
(There are some who dislike starting a sentence with however so some style-guides prohibit it, but there's no grammatical reason not to, and it's as good an option to have available as any other. Most style-guides now allow it.)
We can use however as an aside, pointing out that the sentence (or possibly an independent or parenthetical clause, though as a matter of style that could get fiddly) opposes the previous:
Here we use a comma both before and after.
Finally, we can use it to mean "to whatever extent or degree", "in whatever manner" or "by whatever means". This sense must not use a semicolon or a comma.
If you don't use a semicolon in the cases in your question, you can end up using however in this sense, when you intend the conjunctive adverb sense:
The first sentence here states two separate thoughts, and points out that one is opposed to the other: "Some sentences are ambiguous" and "we try hard to avoid this".
The second sentence states that no matter how hard we try to avoid it, some sentences are ambiguous.
These two grammatically correct sentences differ only in whether we followed the semicolon-and-comma rule you mention, showing its value clearly.
*Four-and-a-half if we include the informal use as an emphatic form of how; "However did you manage that?". This comes from a sense where the original how + ever works, ("How ever did you manage that?") and many would say that this is the only correct form here, and however should not be used.