Apologies are meaningful and important if they are done sincerely and without ego. The ability to accept that you are wrong and to show the personal character to admit your mistake is a big step for some people. It is also important the the receiver of the apology to accept the message and move forward in a positive way.
Sure, apologies mean something, but only if they are sincere and backed up with action. If someone apologizes to me, then turns around and does the same thing again, that apology means nothing. This same fact is true whether the person is a career politician or a member of my family.
Sincere apologies mean a lot. Apologies do mean a lot if the person apologizing is sorry. Society would be better if more people felt sincere about their wrongdoings, and offered heart-felt apologies to those wronged. Unfortunately, some people will learn how to say sorry all of the time and not mean it. This has hurt the credibility of an apology.
Apologies still count. However, it must be sincere and heartfelt. An individual must not only say he is sorry, he must demonstrate it through actions. One that is not sincere does not count, and is easily dismissed. This works with countries, with an apology often going a long way towards healing relations.
You don't do harm and get away with it simply by saying sorry. People who so harm deserve punishment unless they actually feel sorry and guilty and decide to change. Saying sorry does not erase atrocity. Imagine, if Adolf Hitler says sorry and keeps slaughtering, will the Jews pardon him?