When you are talking about the difference between Equality and Fairness, what you are really talking about is Equality vs. Equity.  Fairness is a relative term that is defined by the individual, whereas Equity has been recognized by governmental and educational institutions as a means of upholding the 14th Amendment.

Let me explain:  

Equality means sameness.  Under equality, every person gets exactly the same thing regardless of whether it is needed or not.  Equality only works if everyone starts from the same place.  

Equity on the other hand means "Fairness."  Under equity, everyone has access to the same opportunities.   Equity must be established before Equality can be obtained.

(I am attaching an image that I use in my class to visually represent the difference between the two. *citation below)  Notice in the picture, Equality is providing the same sized box for each child regardless of whether the child needs it or not.  The tallest child does not need the box thus it is a waste of resources.  The smallest child is not receiving enough support, so he is being under-served.  The irony of equality is that when true equality is given equal opportunity is not always received.    

In the illustration, equity provides boxes for the students that need it to obtain the goal of watching the game.  With these added resources, all three children are able to view the baseball game with equal access.  The tallest child however did not receive any accommodations; however, he was able to accomplish the goal without the needed support.  The smallest child had to receive more accommodations than the middle child did.  This does not mean he received preferential treatment, because the goal is Equity rather than Equality.  The smallest child simply needed more help to achieve the same end as the other two children.  

In the United States, the goal is equality; however, some citizens need more support than others.  For example: Immigrants coming into the United States may not speak English fluently; however, the United States grants immigrants the opportunity to become citizens of the United States regardless of their fluency in English.  This establishes the goal of Equality for all potential citizens going through the naturalization process.  However, if the United States government were to use equality in the naturalization process, every candidate would receive an interpreter to read all of the documentation for them as they completed the process.  This would include those candidates who read, write, and speak English fluently.  In addition, in order to achieve equality every candidate would receive exactly the same amount of time with the interpreter- No more and no less.  Both of these examples have the potential to create a state of inequality rather than achieving equality.

Because the United States is Equitable, people who need an interpreter during the naturalization process receive one, while those who do not are allowed to opt out.  Those candidates that have a greater knowledge of the English language may not need the interpreter except to answer a few questions, while those who have very little skill speaking English may need the interpreter's help throughout the entire process.  Under Equity, every candidate for citizenship achieves Equality in their opportunity to become a citizen of the United States.  By offering equitable resources through laws and support agencies the United States can better approach the goal of true equality.  

*Equity image credit: Please note, this image was adapted from an image adapted by the City of Portland, Oregon, Office of Equity and Human Rights from the
original graphic: http://indianfunnypicture.com/img/2013/01/Equality-Doesnt-Means-Justice-Facebook-Pics.jpg

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Treating people equally means that you treat them the same, regardless of their race, sex, social status or anything else like that.  Treating people fairly means that you treat them in ways that are most appropriate to their needs.  This may mean that you treat them differently than you treat others because they have different needs than others do.

A possible example of this (depending on your political point of view) would be affirmative action programs.  These programs do not treat people equally.  Instead, they give preferences to groups that have been discriminated against.  One can argue that these programs treat people fairly because they give benefits to those who have been affected by past discrimination.

If you accept this reasoning, this is a case of treating people fairly without treating them equally.  If you do not accept this reasoning, it is still an attempt to treat people fairly, but one which fails to treat them either fairly or equally.