Let $T(n,c)=\{m\in \mathbb{N} \ : \ m<n, \ \omega(m)<(\log\log n)^c\}$. Then we have $S(n,c)\subset T(n,c)$ and $|S(n,c)|\leq |T(n,c)|$. Also, if $T'(n,c)=\{ m\in \mathbb{N} \ : \ m<n, \ \omega(m)<(\log\log (\sqrt n))^c\}$, we have $ |T'(n,c)|+O(\sqrt n) \leq |S(n,c)| \leq |T(n,c)|$. Thus, we see that if $\lim_{n\rightarrow\infty} |T'(n,c)|/n$ and $\lim_{n\rightarrow \infty} |T(n,c)|/n$ exists and equal, then the limit $\lim_{n\rightarrow\infty} |S(n,c)|/n$ must exist and equal to that of $\lim_{n\rightarrow \infty} |T(n,c)|/n=\lim_{n\rightarrow \infty} |T(n,c)|/n$. As commented above, we use Erdos-Kac Theorem, then we see that $$\lim_{n\rightarrow\infty} |T'(n,c)|/n=\lim_{n\rightarrow \infty} |T(n,c)|/n,$$ and the value of the limit depends on $c$. For $0<c<1$, the value is $0$, For $c=1$, the value is $1/2$, For $1<c$, the value is $1$. Therefore, we obtain that If $0<c<1$, $$\lim_{n\rightarrow\infty} |S(n,c)|/n = 0,$$ If $c=1$, $$\lim_{n\rightarrow\infty} |S(n,c)|/n = \frac12,$$ If $c>1$, $$\lim_{n\rightarrow\infty} |S(n,c)|/n = 1,$$ To obtain a finer result, we use Renyi-Turan (1957) in the form $$ \sup_{x\in\mathbb{R}} \left|\frac1n |\{ m < n : \frac{\omega(m)-\log\log n}{\sqrt{\log\log n}}\leq x\} | - \Phi(x) \right| =O\left( (\log\log n)^{-\frac12}\right). $$ Then we have $$ \left||T(n,c)|/n - \Phi\left( \frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right)\right|=O\left( (\log\log n)^{-\frac12}\right).$$ Then we have the following asymptotic formula $$|T(n,c)|/n = \Phi\left( \frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right) + O\left( (\log\log n)^{-\frac12}\right).$$ To treat the $\Phi$ term, we use Chernoff bound : If $X$ is standard normal distribution, then $$ P(X\geq a) \leq \exp\{\frac{-a^2}2\}.$$ Now, let $\chi(c)$ be $0$, $1/2$, $1$ when $0<c<1$, $c=1$, and $c>1$ respectively. Then we see that $$ |T(n,c)|/n = \chi(c) + O(\exp\{ \frac{-f(n)^2}{2} \}) + O\left( (\log\log n)^{-\frac12}\right) $$ where $$f(n) = \left|\frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right|.$$ We can treat $T'(n,c)$ the similarly.