What is the importance( in Common Law) of the provision of Responsibility distinct from Liability and enabling people to seek the cessation of an illegal act even if they have not been damaged?
The purposes are (1) to prevent damages from materializing (at which point they would become liabilities), and (2) to ascertain who would be liable once damages materialize. Purposes (1) and (2) are known as injunctive relief and declaratory relief, respectively.
Injunctive relief is arguably more drastic and is intended for situations where irreparable loss is imminent or very likely. A respondent's conduct can be of such nature that no award in court could suffice to compensate for the loss caused by that respondent (i.e., the person responsible for the loss); or, if the award compensates at all, the losses would have been exacerbated by the time the correct ruling is made. Personal Protection Orders (example: this one) are one example of injunctive relief.
By contrast, declaratory relief does not strictly require the cessation of an act. Instead, it only determines beforehand or ex-ante the party who would be liable/entitled and the extent of that party's liability/entitlement.
Black's Law Dictionary defines "Responsible" as the entity "[a]ble to pay a sum for which he is or may become liable" (emphasis added). Thus, the concept of responsibility is wider encompassing than the term "liability".