What is the difference between “mourning” and “grieving” someone's death?

It seems to me that both words are interchangeable, I can mourn or grieve the death of a loved one for weeks, months or years. And both terms mean to feel deep sorrow for the loss of someone dear. According to Oxford and Macmillan, to mourn is to feel and manifest sadness in public whereas to grieve is to only feel sadness. Other dictionary definitions do not make this distinction. However, if we look at Oxford's example of usage: “he was mourned for years”, it is nevertheless ambiguous. Did family and friends attend a memorial service every year (public manifestation)? Or did they simply reminisced whenever they met up (in private; i.e. among friends and relatives)? Would one use grieved if it was the latter, or are the two interchangeable?

mourn

Oxford Dictionaries says:

1. Feel or show sorrow for the death of (someone), typically by following conventions such as the wearing of black clothes:
   He was lost in battle to his only enemy, and he was mourned for years after his death by the people who had grown to love him.

The Free Dictionary lists the following three definitions;

The American Heritage Dictionary

1. To feel or express grief or sorrow. See Synonyms at grieve

Collins English Dictionary

1. to feel or express sadness for the death or loss of (someone or something)

Random House Kernerman Webster's College Dictionary

1. to feel or express sorrow or grief.

Longman Dictionary of Contemporary English

1. to feel very sad and to miss someone after they have died [= grieve for]:
   mourn somebody's death/loss/passing
   She still mourns the death of her husband.

Macmillan Dictionary

1. to feel extremely sad because someone has died, and to express this in public
   mourn for: He still mourns for his brother.

grieve

1. Feel intense sorrow:
   _{Oxford Dictionaries}

The Free Dictionary contains the following three definitions:

1. To cause to be sorrowful; distress:
2. To mourn or sorrow for
   _AHD

1. to feel or cause to feel great sorrow or distress, esp at the death of someone
   _CED

1. to feel grief or great sorrow.
   _RHKWCD

1. [intransitive and transitive] to feel extremely sad, especially because someone you love has died
   grieve over/for
   • He died, and every day since then I have grieved for him.
   • People need time to grieve after the death of a loved one.
   • She grieved the loss of her only son.
     _LDOCE

1. [INTRANSITIVE/TRANSITIVE] to feel extremely sad because someone has died

   • I never had time to grieve properly.
   • He’s still grieving for his wife.
   • Millions of people are grieving over his death.
     _MD

According to a blogger called Jade, there is a significant difference between the two terms [emphasis mine].

While grief is the emotional reaction/response to loss, mourning is the process one undertakes to deal with the void that is now left. Mourning is the process of acclimating to living a life without this special someone or something. It is period of adapting to the changes created by this loss.
_{Source: grief and mourning.com}

The following excerpt seems to confirm this difference, grief is the emotional response but mourning is how we process the death of a loved one

Grief is the natural psychological, behavioral, social, and physical response which helps the mourner recognize the loss and get ready for the larger and often longer experience of mourning. As Therese A. Rando says, “Grief is actually the beginning part of mourning.” _{Source: The Difference Between Grief and Mourning}

Q: What is the difference in meaning between the following?

1. People need time to grieve after the death of a loved one
2. People need time to mourn after the death of a loved one

3. She still mourns the death of her husband.
4. She still grieves the death of her husband.

Related: What is the expression for the process of getting over the loss of a loved person?

According to the American Cancer Society the process is grieving, and mourning is a part of it:

Mourning:

• is the outward expression of loss and grief. Mourning includes rituals and other actions that are specific to each person’s culture, personality, and religion. Bereavement and mourning are both part of the grieving process.

Grieving:

• involves many different emotions, actions, and expressions, all of which help the person come to terms with the loss of a loved one. But keep in mind, grief doesn’t look the same for everyone. And, every loss is different.

The following extract appear to confirm the difference between the two terms stated above. In both cases mourning is explained as the "formal" and more public part of the process of grieving:

From Will I grieve or will I mourn? by Alan D. Wolfelt, Ph.D

• Grief is the constellation of internal thoughts and feelings we have when someone we love dies.

  Think of grief as the container. It holds your thoughts, feelings, and images of your experience when someone you love dies. In other words, grief is the internal meaning given to the experience of loss.

• Mourning is when you take the grief you have on the inside and express it outside of yourself.

  Another way of defining mourning is "grief gone public" or "the outward expression of grief." There is no one right or only way to mourn. Talking about the person who died, crying, expressing your thoughts and feelings through art or music, journaling, praying, and celebrating special anniversary dates that held meaning for the person who died are just a few examples of mourning.

Everyone grieves differently from www.helpguide.org:

• Grieving is a personal and highly individual experience. How you grieve depends on many factors, including your personality and coping style, your life experience, your faith, and the nature of the loss. The grieving process takes time. Healing happens gradually; it can’t be forced or hurried—and there is no “normal” timetable for grieving. Some people start to feel better in weeks or months. For others, the grieving process is measured in years. Whatever your grief experience, it’s important to be patient with yourself and allow the process to naturally unfold.