Are these two regular expressions equivalent?
First, an alternative notation (what I usually use, and it is my personal preference) to join two regular expressions is to use '+' instead of $\cup$ which is used to denote a union of two sets. So your regular expression can be written as 
$$ u^*v(w + xu^*w)^* $$
or sometimes
$$ u^*v(w \mid xu^*w)^* $$
instead of
$$ u^*v(w \cup xu^*w)^* $$
That pair of parentheses around the second $u$ is redundant and so not needed since the Kleene star operator applies to only $u$ in $xu^*w$, i.e., $xu^*w$ and $x(u)^*w$ denote the same set.