To "like something" is to express positive feelings towards it, they may not be particularly strong feelings but it is not a neutral state of being. To say "I don't like something" is to express the opposite viewpoint and hence you express a view that is negative towards it, again it may not involve strong feelings but it is taken to be a negative.
It's part of the limitations of language - if you wish to express a neutral viewpoint then say "I've got no feelings about it", "I don't care about it either way", "I'm utterly indifferent to it" but "don't like" is normally taken as an antonym of "like".
Technically, to say that you don't like something, just means that you don't like it. It could cover anything on the spectrum of liking that isn't an active liking. To dislike something would be to actively feel negatively towards it.
However, in practise, "don't like" and "dislike" are used interchangeably.