I think your question contains its own answer. Liking something and disliking something are both active things. From the OED:
Like: Find agreeable, enjoyable, or satisfactory
Dislike: Feel distaste for or hostility towards.
To "not like" something would, strictly speaking, imply merely the absence of the agreeable or pleasant reaction that liking something has, but without the distaste or hostility implicit in disliking something.
That said, I'm sure that some, indeed many, people use "don't like" as a euphemism to "soft pedal" the fact that they do actually dislike something. And that they probably do it without conscious thought.
When I think of my work colleagues there are many that I "don't like" but don't dislike either because I really don't know them. But I can think of a handful who I do actively dislike. I probably wouldn't use either term publicly, if only to keep the peace, but if push came to shove I'd more likely use the former rather than the latter for the same reason. Even, perhaps especially, for the one that I don't merely dislike but despise.
As another example (with a nod to your automotive avatar) I "don't like" the Ford Falcon. I don't dislike it either. It evokes no feelings in me one way or the other. But I "disliked" the Holden Astra, because it had as much driver visibility as a Type VII U Boat. You don't so much drive one of those things as "use the Force and hope for the best", so there was a reason to actively dislike it.
In short, while I may sometimes use "don't like" as a euphemism for "dislike", I don't use "dislike" as a synonym for "don't like" but rather reserve the word for when I have an active negative feeling about something. I'm sure I'm not the only one who does that.