Not much.
Since in Bitwise Sieve we use the individual bits to store whether the $n^{th}$ number is prime or not, only the memory requirement is reduced to ${1/8}^{th}$ versus the normal Sieve of Eratosthenes, where we use a boolean array to do the same. This however, does not improve the time complexity, its still $O(N\ log\ log\ N)$.
Almost all of the online judges provide enough memory to make a boolean array of size upto $10^8$ i.e. ~**100 MB**. So, most of the problems can be solved using the normal Sieve. (including [TDPRIMES][1]). Bitwise Sieve is only required when the memory limit (**ML**) is kept deliberately low so that the naive solutions don't pass.
For example: [https://open.kattis.com/problems/primesieve][2]. Notice that the **ML** is only **64 MB**.
So, in general, Bitwise Seive is not of much ~~important ~~importance in contests.
[1]: http://www.spoj.com/problems/TDPRIMES/
[2]: https://open.kattis.com/problems/primesieve