How should I connect this dependent clause after a conjunction

Let's consider the simplified sentence

[1] Almost all calculus is taught using methods that were appropriate for the era of paper and pencil but not for the computer era.

Summary

Neither a colon nor a semi-colon would work here. All of the rest would work: but not (with or without a preceding comma), a bare not (preceded by a comma), or an em-dash. Probably an em-dash is a bit much—unless the writer as a good reason for the added emphasis.

Not a dependent clause

Grammatically, but not for the computer era is not a dependent clause. Rather,

[2] for the era of paper and pencil but not for the computer era

is a subclausal coordination of two preposition phrases, namely for the era of paper and pencil and for the computer era. In [2], but is a marker of coordination, while not is a modifier of the second coordinate. (This is because not for the computer era is not itself a preposition phrase; we can't have almost all calculus is taught using methods that were appropriate not for the computer era.) See CGEL, p. 1313, example [60 ii a.] They had invited Jill but not her husband.

Comma before but: optional

[3] Almost all calculus is taught using methods that were appropriate for the era of paper and pencil, but not for the computer era.

It is a matter of style whether a comma is placed before the but. Commas are often optional; 'a light style puts in relatively few commas (or other marks) in those places where they are optional rather than obligatory' (CGEL, p. 1727).

Comma+not on their own: acceptable

The following is acceptable (CGEL, p. 1313):

[4] Almost all calculus is taught using methods that were appropriate for the era of paper and pencil, not for the computer era.

While this is also coordination, it doesn't use an explicit coordinator (i.e. it is asyndetic coordination). CGEL emphasizes that the versions with and without the explicit but are generally not equivalent: if we say They had invited Jill but not her husband, 'we understand that they might in principle have invited both Jill and her husband, but in fact did not do so'. In contrast, if we say They had invited Jill, not her husband, then 'the issue is which of Jill and her husband it was that they invited'. However, in our case, [1] and [4] have the same meaning.

Semicolon on its own: not acceptable

Normally, a semi-colon separated complete clauses, which is not what we have here. True, a semi-colon is sometimes appropriate as a mark of subclausal coordination. But this happens only in relatively formal style when the coordinates are long and/or complex, and especially when at least one of the coordinates has an internal comma (CGEL p. 1740). That is not in our case.

Colon on its own: not acceptable

Let's consider

[5] ^*Almost all calculus is taught using methods that were appropriate for the era of paper and pencil: not for the computer era.

I agree with Yosef Baskin that a colon doesn't work here (which is why it is preceded by an asterisk).

In a sentence such as [5], the colon would be interpreted as marking asyndetic subclausal supplementation. In other words, in [5], not for the computer era is supposed to function as a supplement.

A supplement is a part of a sentence that isn't syntactically integrated to the rest of the sentence. Instead, it is only semantically related to another part of the sentence, the so-called anchor.

Now, according to CGEL (p. 1741), a 'colon is admissible [as a marker of supplementation] only if the supplement follows the clause containing the anchor,' and, moreover, the supplement must be providing identifying rather than merely descriptive information.

To illustrate this last point, consider

[6] They went to Bill Clinton, the only man who could help them.

CGEL says that 'a colon would be out of order in' [6], 'because the supplement provides descriptive,not identifying, information - compare They went to the only man who could help them: Bill Clinton, where the supplement does identify.'

Now let's go back to our sentence [5]. The form of the would-be supplement is a modifier (not) + a preposition phrase (for the computer era). It would be most naturally interpreted as a predicative complement (as in, this is [not for the computer era]). This means that the anchor in the main clause would have to be something semantically compatible with a predictive complement (for a detailed discussion of this point, see the answer). The anchor would therefore have to be a noun phrase, arguably this one:

[7] methods that were appropriate for the era of paper and pencil

On the one hand, we see that this noun phrase is indeed semantically compatible with the supplement. On the other hand, however, the supplement is not identifying; not for the computer era is just elaborative and descriptive of [7]; it does not serve to identify it. Therefore the colon is not appropriate here.

An em-dash: acceptable

[8] Almost all calculus is taught using methods that were appropriate for the era of paper and pencil—not for the computer era.

Here the em-dash serves to 'set off some constituent from the rest of the text' and 'provides an elaboration, explanation, or qualification of what precedes.' Thus, [8] is most closely related to [4], except that not for the computer era is more strongly set off and thus is emphasized.

The stylistic winner

This will depend on what one is trying to do. In the comments, Yosef Baskin says that an em-dash is an overkill. That's probably right, but I can imagine circumstances where the writer might want the additional emphasis even in this case. As far the other acceptable possibilities, which one we choose is really just a matter of taste, except that the presence or absence of the comma before but should be consistent with the rest of the text: present if the punctuation style is heavy, absent if light.