The difference is pretty basic: Having a "negative impact" requires value-judgement, having a cost does not. In terms of economics it requires a broad view of the entire economy, whereas a cost can be directly assigned to a single event or series of events.

Saying that a regulation has a cost is saying that it requires some expenditure to implement/maintain. Saying it has a negative impact is saying that the expenditure to implement/maintain that regulation outweighs the as-yet-in-this-discussion undefined benefits of the regulation in question.