Out of these two algorithms. Is there always an input where A is faster then B? (Big theta notation) I am currently learning landau notations and am stuck on the following True/False question.



What seems a little confusing to me is the use of big-theta notation to describe worst-case run-time. 
What is the difference in saying "worst-case run-time Θ(n log n)" or simply "run-time O(n log n)"? Giving the worst case a tight limit seems a lot like simply giving an upper bound to the run-time.

If I go by that reasoning my guess would be that the statement is true. As an O(n) algorithm will be faster than an O(n log n) as n grows large enough.
But i am not sure if my reasoning is correct and if i get the big picture here.
Θ(n log n) means the algorithm is O(n log n) and Ω(n log n), and is slower than Θ(n) so your reasoning is correct. 

But there's another layer: Landau notation denotes asymptotic bounds on functions. Saying "worst case" restricts it to those functions which result in the highest run time. For example take binary search: we get a best-case runtime asymptotic of Θ(1) and a worst-case asymptotic of Θ(log n). 

To answer your question about the difference between "worst-case Θ(n log n)" and "run-time O(n log n)"*, the first one gives us more information: it tells us it runs about as fast as n log n, where the second tells us it runs at most n log n.

*Technically the second one doesn't tell us for what 'run-time' configuration it is asymptotically approaching, but in that case generally most people mean 'in the worst case this algorithm runs in O(n log n)' when they don't say anything.