What is the relationship between internal audit and quality assurance and what are the similarities / differences between them?

The Internal Audit is one of the requiremnts of Quality Assurance.
an Internal audit is a tool used for organisation for self assurance that its process are effective and efficient.
The output of internal audit is usually used to improve other quality assurance procedures such as lack of control procedures or more effective control procedure or lack of resources...etc.

Internal audit has wider scope than quality assurance,internal audit includes all activities inside control environment,however they both have same objective.The similarities may be in the procedures applied and the main difference is in scope.
Hope it was useful.

Here is a link to an article that deals with this issue:

http://smallbusiness.chron.com/quality-assurance-vs-audit-17455.html