The difference between apology and sorry

This post merely express my own understanding
about the differences between apology and sorry.

Let us look at the definition in some of the dictionaries.
I refer to Oxford Advanced Learner's Dictionary (OALD),
Cobuild Dictionary and Merriam-Webster (M-W) Dictionary.

OALD defines apology as:
- A word or statement saying sorry for something that
  has been done wrong or that causes a problem

Cobuild defines apology as:
- An apology is something that you say or write in order to tell someone
  that you are sorry that you have hurt them or caused trouble for them.

Cobuild illustrated the definition stated above with the following phrase:
If you say that you make no apologies for what you have done,
you are emphasizing that you feel that you have done nothing wrong.

M-W defines apology as:

- An admission of error or discourtesy accompanied by an expression of regret  
  <a public apology>

Example: The company issued an official apology to its customers for its error. 

Hence, it is obvious that making apologies,
one actually admits the wrong doing(s) which caused hurt or
trouble(s) in the event and the person acknowledges the accountability.

On the other hand, expressing sorry is different from apology.

OALD defines sorry as:
- Feeling sad and sympathetic, or
- Feeling sad and ashamed about something that has been done
- Feeling disappointed about something and wishing you had
  done something different or had not done something

Cobuild defines sorry as:
- You say `Sorry' or `I'm sorry' as a way of apologizing to someone
  for something that you have done which has upset them or caused them difficulties,
  or when you bump into them accidentally.
- If you are sorry about a situation,
  you feel regret, sadness, or disappointment about it.
- You use 'I'm sorry' or 'sorry' as an introduction when you are telling
   a person something that you do not think they will want to hear,
   for example when you are disagreeing with them or giving them bad news.

M-W defines apology as:
- Feeling sorrow, regret, or penitence

It seems that expressing sorry only shows emotion,
regret and even disagreement towards the event or others.
The admission of error and/or fault is not in place.

Very often we hear about certain nation and/or politician express
sorry about certain event took place but there is no admission
of error and/or fault and certainly does not acknowledge accountability.

Another interesting phenomenon is customer service or service sector 
like to express sorry whenever there is unhappiness in an unpleasant situation. 

Some trainers from the old school will teach employees and/or
staff to express sorry towards the clients and/or customers
and continue to do so in the trainings and workshops.

I think it is time for the trainers and the respective personnel
in the customer service to realise the differences between the two.

Customers are not looking for the organisations' regret or sympathy in their response,
if a representative and/or organisation merely express sorry on an event or situation,
is there any real good or purpose expressing sorry?
At least there should be decency to admit the error(s) and/or wrongdoing(s).

If organisations are not sincere and not ready to admit their mistakes, 
it merely demonstrate the corporate culture of not taking responsibilities.
In the end, it will ruin the brand as well as image of the organisation.