Should I use a semicolon or a dash to connect two closely related sentences?

I don't think I was ever taught a clear-cut rule, and as a non-native speaker, I am probably spoiled to some extent by the usage of dashes in other languages. That being said, following nothing but my intuition I would use:

• a semicolon when the sentences express related, yet independent (especially grammatically independent) thoughts; they could well stand on their own, separated by a period.
  
  
• a dash when the second sentence backs up the first one, nails it down to something, restates or amplifies it, provides reasons or examples, or when the second sentence could not stand on its own "as is" for grammatical reasons.

I will try to demonstrate my point by rewording the notorious examples from The Oatmeal accordingly.

• My aunt had hairy knuckles; she loved to wash and comb them.
• My aunt had hairy knuckles — she suffered from hirsutism.
  
  
• When dinosaurs agree on something, they'll often high-five one another; dinosaurs are all about high-fives.
• When dinosaurs agree on something, they'll often high-five one another — they cannot talk and have to resort to gestures.
  
  
• I gnaw on old car tires; it strengthens my jaw so I'll be better conditioned for bear combat.
• I gnaw on old car tires — to strengthen my jaw so I'll be better conditioned for bear combat.

Again, this is just my two cents, and I'm only putting them in because the other answers so far seem to miss the point of your question by focusing on non-sentential relations and sometimes not even mentioning semicolons at all. I don't know whether my answer comes close to being correct or not, but I hope it will at least serve as a turning point for getting the discussion back on topic.

Semicolon is used to join sentences that can stand alone, but are joined to emphasize their relationship.

En dashes are used to indicate periods of time or other numerical ranges.

Hyphens are used to combine open compounds.

Em dashes are used to disrupt the flow of a sentence and bring emphasis to the coming point. It is a more informal and stronger version of the colon. It can also act as a stronger version of the comma.

Typically, if I'm connecting two closely related sentences, I use a semicolon; I use dashes in instances where commas would also be acceptable—usually where commas would be confusing—or where parentheses would also be acceptable.

Firstly, you should be aware that there are two different kinds of dashes: the en dash and the em dash.

The em dash (—; historically, an em is precisely the width of the letter "M", but now defined by the height of the font) is generally used to add a passage into the middle or end of a sentence. This is similar to using parenthesis, but should be read without adding a pause. Moreover, the em dash in fact interrupts the sentence, so no pause should be used at all; an interruption should be emphasized.

This dash is also often used as a de facto interruption; for example in dialog when the speaker for some reason cannot continue.

In use, the em dash is not surrounded by spaces. This is done—like this—because it otherwise uses excessive spacing.

Some publishers or style guidelines may instead prefer the narrower en dash (–; an en half the width of an em). This character is usually – as in this example – surrounded by spaces.

Suggestions on the use of semicolons can be found as answers to this question.