Q:

What does ranking up do?

When I collect a card of a class I already have, its rank goes up by one, but what exactly does this rank do?  For instance, how does a human vanguard - III compare to the human vanguard - I?

A:

Ranking up a class card grants you different appearance options, and experience.  You are also given skill points.  As someone pointed out in the link below, even ranks give you skill points, while odd ranks given you appearance cosmetics.
There is also a way to flip the card over to see what each rank does according to this source:

When you flip the card, you can press another button to check what ranks do. Every even rank gives you skill points while every odd rank gives you armor tints or patterns.

Although they didn't state which button, the screen likely displays which button it is.  One source states you can get a total of 25 skill points.
I've also found an additional source stating the same things.

A:

At rank 1, a level 1 character has 

3 spent skill points (1 in each active ability)
2 unspent skill points
color tint 1

Rank increases each time you get that character's card from loot boxes.
Odd ranks unlock appearance options:
rank 3 : color tint 2
rank 5 : color tint 3
rank 7 : pattern select
rank 9 : pattern color

Even ranks give skill points:
rank 2 : +4 skill points
rank 4 : +4 skill points
rank 6 : +5 skill points
rank 8 : +5 skill points
rank 10 : +5 skill points

At rank 10, a level 1 character has

3 spent skill points (1 in each active ability)\
25 unspent skill points

As a character levels, additional skill points are earned.  These skill points are lost if the character is prestige'd (reset to level 1).  However, skill points from ranks are not lost.  This creates a situation where you cannot look at only a character's level to evaluate available skill points.  A rank 10 level 1 character will be more powerful than a rank 1 level 5 character.

