Is it not due to the formations i.e in the classic 4-4-2 you have 2 centre backs and 2 fullbacks (left back and right back). And for a more defensive formation i.e. 5-3-2-1 you have 5 backs - which the 2 on the ends are known as wingbacks.
They are generally differentiated by the system they play in and the demands these systems place on them.
System:
The modern wingback will be found in any system with 3 centrebacks such as a 3-4-3; 3-5-2 or 5-4-1.
A modern fullback will be found in your 4-man backline. This can be in a multitude of formations such as the 4-4-2; 4-4-3; 4-5-1; 4-2-3-1 etc.
The one exception here is that you can play with a 5-man defensive line and use fullbacks and not wingbacks.
Positioning and Roles:
The wingback tends to sit further up the pitch than the fullback and they also have greater responsibility. When you play a wingback you don't have wide attacking players (no LW/RW or LM/RM) which means not only does the wingback have to defend the wide areas he also has to provide his team's attacking threat from out wide. This means he has to make runs down the wings to provide crosses and then defend against the opposition's wide attacking players.
The fullback on the other hand sits deeper than the wingback and tends to work with a wide midfielder (LM/RM) or winger (LW/RW), so they share the responsibilities of attacking and defending. If the fullback attacks, the wide midfielder/winger will sit back and provide defensive cover.