Let $\omega(n)$ denote the number of distinct prime factors of $n$ and $\Phi(\cdot)$ denote the Normal distribution with mean $0$ and variance $1$. Then, uniformly in $t$, the number of integers $n \leq x$ with $\omega(n) \leq \log\log n + t \sqrt{\log\log n}$ is $$\Phi(t) + O\left(\frac{1}{\sqrt{\log\log x}}\right)$$ as $x \rightarrow \infty$. ** The error term is sharp. **

This particular error term was conjectured to hold by Leveque in the 40's. His conjecture was settled a few years later by Erdos and Renyi.

Before Erdos and Renyi's paper the best error term was $O(\log\log\log x / \sqrt{\log\log x})$ and was due (if I recall correctly) to Kubilius. Kubilius's method was in its origin probabilistic and relied in an essential way on truncating the "random variable" $\omega(n)$. This introduced an additional factor of $\log\log\log x$ to the error term, as inevitably, truncation leads to loss of information.

In contrast, Renyi's and Erdos's method was purely analytic: the idea was to estimate $\sum_{n \leq x} exp(\text{i}t \omega(n))$ uniformly in $t$ in a certain range, and extract the desired conclusion from the behavior of this sum. To this end, they apply Berry-Esseen's theorem, but in principle one could use an more hands-on approach: smooth the indicator function of $\omega(n) \leq \log\log n + t \sqrt{\log\log n}$ and express it in terms of a variant of Perron's formula, after summing the resulting expression over $n \leq x$ one could proceed with the saddle-point method.

However when purely analytic methods are not directly accessible, Kubilius's method is the canonical method. For example suppose that you want to investigate the distribution of $\omega(n)$ over a peculiar subset of $[1,x]$ on which sieve methods -- but not "heavy" analytic methods -- are applicable, then Kubilius's method is still your best bet.

(As far as terminology is concerned it's "Kubilius model's" rather than "Kubilius's method"; more details about "Kubilius's model" can be found in Volume 1 of Elliott's "Probabilistic Number Theory").