The notation $a:b$ emphasizes a relative relationship between $a$ and $b$, and the notation $\frac{a}{b}$ emphasizes an operation on two elements $a$ and $b$.
But ultimately the two symbols represent the same thing (at least, when using integers): a relative size between $a$ and $b$. If you think carefully about what it means for two ratios to be equivalent, you'll find that the definition of equality of $\frac{a}{b}$ and $\frac{c}{d}$ is just that the ratios $a:b$ and $c:d$ are equal.
Actually, I do believe that I've seen posters from certain countries actually use "$:$" for division, to the confusion of the rest of us.
One difference in these two notations is that you can link a lot of ratios together at once like this: $1:2:4:7$. This expresses a bunch of ratios at once: $1:2$, $2:4$, $4:7$, $1:4$, $2:7$ etc. If these were ratios of ingredients in some mixture recipe, then you could rather handily increase and decrease the size of your recipe as you desired using this notation.
But this does not translate over to the slash notation, which becomes problematic if you're thinking of the slash as an operation.
This is a bit of a reach, but one way to think of it is that $a:b$ is kind of like "a division operation you are postponing." This is why you can stack them together because no operation is intended. (If you used slashes, the urge would be to carry out the operations until you have a single fraction, but this would require parentheses to make the expression unambiguous.)
The length of a continued fraction makes a fraction more (subjectively) complex.
All rational numbers can be written as a finite continued fraction, as there is an algorithm that computes continued fractions, but only stops if the number is rational.
There is evidence to suggest that certain cultures find rational numbers with simple continued fraction representations more tuning. In Western music, the preferred interval for the minor third is $6:5$, or $[1; 5]$ in continued fraction notation. However, this interval can also be tuned as $32:27$, or $[1; 5, 2, 2]$. The first interval clearly has a more compact representation, so to Western ears, it sounds "nicer". (Wikipedia)
When comparing two continued fractions of the same length, the fraction with a larger final term might be considered "uglier" as it is closer to a rational approximation with fewer terms.
On the contrary, as Gerry Myerson suggests, fractions with large denominators might have short continued fraction representations and vice versa. This suggests there needs to be a trade-off between the length of the continued fraction and the size of the denominator.
Best Answer
Making this an answer: You would do the same thing with scientific notation that you would do with integers or any other reals. $$ \dfrac {a \cdot 10^p}{b \cdot 10^q} = \dfrac {a}{b} \cdot 10^{p-q}. $$Here, we have $$ \dfrac {2.69 \cdot 10^{-8}}{2.23 \cdot 10^{-7}} = \dfrac {2.69}{2.23} \cdot 10^{-1} = \dfrac {269}{2230}. $$