Big-O is an upper bound.
Big-Theta is a tight bound, i.e. upper and lower bound.
When people only worry about what's the worst that can happen, big-O is sufficient; i.e. it says that "it can't get much worse than this". The tighter the bound the better, of course, but a tight bound isn't always easy to compute.
See also
Related questions
- What is the difference between Θ(n) and O(n)?
The following quote from Wikipedia also sheds some light:
Informally, especially in computer science, the Big O notation often is
permitted to be somewhat abused to describe an asymptotic tight bound
where using Big Theta notation might be more factually appropriate in a
given context.
For example, when considering a function T(n) = 73n3+ 22n2+ 58, all of the following are generally acceptable, but tightness of bound (i.e., bullets 2 and 3 below) are usually strongly preferred over laxness of bound (i.e., bullet 1
below).
T(n) = O(n100), which is identical to T(n) ∈ O(n100)T(n) = O(n3), which is identical to T(n) ∈ O(n3)T(n) = Θ(n3), which is identical to T(n) ∈ Θ(n3)
The equivalent English statements are respectively:
T(n) grows asymptotically no faster than n100T(n) grows asymptotically no faster than n3T(n) grows asymptotically as fast as n3.
So while all three statements are true, progressively more information is contained in
each. In some fields, however, the Big O notation (bullets number 2 in the lists above)
would be used more commonly than the Big Theta notation (bullets number 3 in the
lists above) because functions that grow more slowly are more desirable.
