Learn English – What’s the difference between “adviser” and “advisor” — are both interchangeable

In the simplest terms, proof is conclusive but evidence isn't. Evidence is more of a suggestion.

In your example sentence, the use of evidence is to make it clear that there are other possibilities, such as maybe the dating being wrong, or the evidence not being human in origin.

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.