What is the difference between an indexed family and a sequence? For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J
For sequence wikipedia states: Formally, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
Both can have repeated elements and their order matters. Is there any difference between this two?

You can index something by the real numbers for example.  So for every real number $\alpha$ you might have a value $x_{\alpha}$.  But since you cannot enumerate the real numbers you cannot represent that as a sequence.  So it's just an indexed set, something more general than a sequence.
