Ah, I see! The rule is: use aussi with adjectives and adverbs and autant with verbs.

Je chante aussi bien que ma tante. -- I sing as well as my aunt.
Je chante autant que ma tante. -- I sing as much as my aunt.

In the first sentence the focus is on how well I sing in comparison to my aunt. The second sentence is about how much I sing. Same in your second example:

Oui, elle nage autant que Laure.  -- Yes, she swims as much as Laure.
Oui, elle nage aussi vite que Laure. -- Yes, she swims as fast as Laure.

I hope this helps a bit.

Je chante aussi bien que ma tante

Ils voyagent autant que nous. (Different example)

And here is « nage » example I keep getting wrong on test because I think it should be aussi: