How can $\Theta$ and $O$ complexities be different?
If the $O$-complexity given is true, then $f(x)\leq c_2n\log n$ will not always be true. So in this case, how is $\Theta$-complexity different from $O$-complexity?


Yuval has covered the quicksort aspects of your question but you have a couple of fundamental misunderstandings about asymptotics.

There is no such thing as "$\Theta$-complexity" or "$O$-complexity". Asymptotic notations such as $\Theta$ and&nbsp;$O$ are simply ways of describing the growth rate of functions. Those functions could be used to measure anything at all (including nothing at all). The complexity is the function (or, rather, the running time is measured by the function); the asymptotic notation is just a way of describing that function's behaviour.  As an analogy, suppose that I tell you that my height is "less than two metres": you wouldn't say that my "less-than height is two metres". Likewise, if I say that I weigh about 70kg, you wouldn't say that my "about weight is 70kg." "Less than" and "about" are just ways of describing the numbers that measure those physical properties.

The second misconception is that, for a given function&nbsp;$f$ there is some unique function&nbsp;$g$ such that it's correct to write $f=O(g)$. Remember that $f=O(g)$ means (in a way that's more precise than the following description) "$f$&nbsp;is kinda less than&nbsp;$g$."  To continue the analogy of the previous paragraph, it's completely true to say that my height is less than 2m and that my height is less than 102m.  Indeed, there's also no contradiction in saying "My height is about 1.8m and less than 2m", which is directly analogous to "The running time is $\Theta(n\log n)$ and $O(n^2)$."