Asymptotically, the proportion of semiprimes $\leqslant N$ where the two factors have roughly the same size is negligible, if we say that two primes $p \leqslant q$ have roughly the same size if there is a constant $\alpha > 1$ such that $q \leqslant p^\alpha$.
The number of semiprimes $\leqslant N$ is
$$\pi_{1/2}(N) = \sum_{p_k \leqslant \sqrt{N}} \left(\pi\left(\frac{N}{p_k}\right) - (k-1)\right),\tag{1}$$
and the number of semiprimes with two factors of $\alpha$-roughly the same size is
$$\pi_{1/2;\,\alpha}(N) = \sum_{p_k \leqslant N^{1/(\alpha+1)}} \left(\pi(p_k^\alpha) - (k-1)\right) + \sum_{N^{1/(\alpha+1)} \leqslant p_k \leqslant \sqrt{N}} \left(\pi\left(\frac{N}{p_k}\right) - (k-1)\right).\tag{2}$$
The sum of the subtracted terms is the same in both,
$$\frac{\pi(\sqrt{N})(\pi(\sqrt{N}-1)}{2} \sim \frac{2N}{(\log N)^2}.$$
In the first sum of $(2)$, we can estimate the contribution of the $\pi(p_k^\alpha)$ as
$$\sum_{p_k \leqslant N^{1/(\alpha+1)}} \pi(p_k^\alpha) \leqslant \pi(N^{1/(\alpha+1)})\cdot \pi(N^{\alpha/(\alpha+1)}) \sim \frac{(\alpha+1)^2 N}{\alpha(\log N)^2},$$
so it remains to consider
$$\sum_{A \leqslant p_k \leqslant B} \pi\left(\frac{N}{p_k}\right),\tag{3}$$
where here $B = \sqrt{N}$, and $A$ is either $2$ (for $(1)$), or $N^{1/(\alpha+1)}$ (for $(2)$).
To obtain the asymptotic behaviour, we don't need a too good approximation, $\pi(x) \sim x/\log x$ is sufficient. So
$$\begin{align}
\sum_{A \leqslant p_k \leqslant B} \pi\left(\frac{N}{p_k}\right)
&\approx \sum_{A\leqslant p_k\leqslant B} \frac{N}{p_k(\log N - \log p_k)}\\
&= \frac{N}{\log N} \sum_{A\leqslant p_k\leqslant B} \frac{1}{p_k\left(1 - \frac{\log p_k}{\log N}\right)}.
\end{align}$$
We have $0 < \log p_k \leqslant \frac12\log N$, and for $0 \leqslant x \leqslant \frac12$, we have $1+x \leqslant \frac{1}{1-x}\leqslant 1+x+2x^2$. Using further $p_k \sim k\cdot \log k$, the major term in the sum is
$$\begin{align}
\sum_{A \leqslant p_k \leqslant B} \frac{1}{p_k} &\sim \sum_{A\leqslant p_k\leqslant B} \frac{1}{k\log k}\\
&\sim \log \log \pi(B) - \log \log \pi(A).
\end{align}$$
For $(1)$, the sum is $\log \log N + O(1)$, while for $(2)$, the sum is $\log \frac{\alpha+1}{2} + O(1)$. For the next largest term, we get
$$\frac{1}{\log N}\sum_{A\leqslant p_k\leqslant B} \frac{\log p_k}{p_k} \approx \frac{1}{\log N}\sum_{A\leqslant p_k\leqslant B} \frac{1}{k} \approx \frac{\log (\pi(B)/\pi(A))}{\log N},$$
which is bounded, and similar for the $x^2$ term. So we have
$$\pi_{1/2}(N) \sim \frac{N\log \log N}{\log N},\tag{4}$$
and
$$\pi_{1/2;\,\alpha}(N) \sim c_\alpha\frac{N}{\log N} \in \Theta\left(\frac{N}{\log N}\right).\tag{5}$$
With more careful estimates, one can determine the constant $c_\alpha$, but the above is sufficient to see that asymptotically, almost all semiprimes $\leqslant N$ have factors of very different size.
However, the factor $\log \log N$ grows very slowly, so one needs to consider the semiprimes up to a really large limit for the proportion of semiprimes with two close factors to become negligible.
Obviously, I have neither the time nor the resources to compute the proportion for really large numbers, but the counts for small limits ($\alpha = 2$) already show a trend:
Limit Primes Semiprimes Close semiprimes
100: 25 (1.1513) 34 (1.0253) 14 (0.6447; 41.1765)
200: 46 (1.2186) 62 (0.9851) 22 (0.5828; 35.4839)
500: 95 (1.1808) 153 (1.0409) 47 (0.5842; 30.7190)
1000: 168 (1.1605) 299 (1.0687) 87 (0.6010; 29.0970)
2000: 303 (1.1515) 577 (1.0811) 167 (0.6347; 28.9428)
5000: 669 (1.1396) 1365 (1.0855) 377 (0.6422; 27.6190)
10000: 1229 (1.1320) 2625 (1.0889) 693 (0.6383; 26.4000)
20000: 2262 (1.1201) 5081 (1.0973) 1317 (0.6521; 25.9201)
50000: 5133 (1.1108) 12110 (1.1004) 3115 (0.6741; 25.7225)
100000: 9592 (1.1043) 23378 (1.1015) 5938 (0.6836; 25.3999)
200000: 17984 (1.0976) 45230 (1.1033) 11339 (0.6920; 25.0696)
500000: 41538 (1.0902) 108326 (1.1044) 26616 (0.6985; 24.5703)
1000000: 78498 (1.0845) 210035 (1.1051) 50983 (0.7044; 24.2736)
2000000: 148933 (1.0804) 407284 (1.1046) 97052 (0.7040; 23.8291)
5000000: 348513 (1.0752) 979274 (1.1042) 230514 (0.7111; 23.5393)
10000000: 664579 (1.0712) 1904324 (1.1041) 442445 (0.7131; 23.2337)
20000000: 1270607 (1.0680) 3704340 (1.1034) 851642 (0.7159; 22.9904)
50000000: 3001134 (1.0641) 8940570 (1.1025) 2024347 (0.7177; 22.6423)
100000000: 5761455 (1.0613) 17427258 (1.1019) 3904631 (0.7193; 22.4053)
200000000: 11078937 (1.0588) 33992717 (1.1011) 7534855 (0.7201; 22.1661)
The numbers in parentheses are
- $\dfrac{\pi(x)\log x}{x}$,
- $\dfrac{\pi_{1/2}(x)\log x}{x\log\log x}$,
- $\dfrac{\pi_{1/2;\,2}(x)\log x}{x}$,
- $\dfrac{100\cdot\pi_{1/2;\,2}(x)}{\pi_{1/2}(x)}$.
We see that $(4)$ is a decent asymptotic even for smallish numbers, while $(5)$ still has considerable fluctuation in the checked range, but we see that the proportion of semiprimes with two close factors drops steadily.
