Punctuation – Does a Comma Before ‘When’ Change Its Meaning?

The comma in question separates coordinate clauses. 

It's not the conjunctions themselves that require this comma.  It's the compound sentence structure that requires both a comma and a coordinating conjunction.  The words "and", "but" and "or" function only as coordinating conjunctions, so compound sentences joined by them will always* include the comma. 

The words "so", "for" and "yet" sometimes function as coordinating conjunctions, but they also have other functions.  "So" is an interesting word because it sometimes functions as a subordinating conjunction. 

Your example sentence uses "so" as a subordinating conjunction. 

With the comma, we have a compound sentence: 

Exercise causes blood vessels to open up, so blood flows more easily. 

Here, the two clauses are independent.  Each clause could stand as a separate sentence. 

Without the comma, we have a complex sentence: 

Exercise causes blood vessels to open up so blood flows more easily. 

Here, "blood flows more easily" is subordinate.  The phrase "so blood flows more easily" modifies the phrase "to open up". 

There is practically no difference in meaning between the complex and compound versions of that sentence.  The difference might be easier to see given a different example: 

He's throwing a party so I can meet her. 
He's throwing a party, so I can meet her. 

In the first, we know the reason that he's having a party.  In the second, we have no idea. 

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* This "always" isn't quite true.  Some style guides suggest that short, well-balanced clauses don't need this comma.  It's confusing but it's common.** 

** The clauses "it's confusing" and "it's common" are short and well balanced.

As it's written, your sentence is the equivalent of separating two independent clauses with a comma. That's known as a comma splice. While a comma splice is acceptable in some situations, I wouldn't say this is one of them.

There are multiple ways you can deal with this.

1. You can explicitly separate the sentence into two independent clauses with different punctuation:

Your dad's bought a car. That's great.
Your dad's bought a car; that's great.
Your dad's bought a car—that's great.

2. You can turn the second part of the sentence into a clarifying term for the first part with a colon:

Your dad's bought a car: that's great.

3. You can use a comma followed by a conjunction:

Your dad's bought a car, and that's great.

4. You can use a comma but turn what follows it into a nonrestrictive clause:

Your dad's bought a car, which is great.

Note that this leaves the meaning of the sentence ambiguous. It's not clear if it's the fact that he bought a car that's great or if it's the car itself that's great.

5. You can eschew punctuation altogether but make what currently follows it into a restrictive clause:

Your dad's bought a car that's great.

Here, it's clear that it's the car itself that's great, not his purchase of it.

Some of these variations are simply a matter of preference, while others are determined by the exact meaning of what you're trying to convey.