Shortest answer: Because mathematics is useful.
Short answer: I see no reason to use a unified notation just because there exists some highly theoretical framework in which two things are identical.
The purpose of mathematical notation is an optimal way of conveying information from the author to the reader. Since people generally start to learn sums and integrals long before they learn measure theory, it is (at that point) completely natural that two notations are introduced.
Later, when the two things are shown to be two sides of one coin, you can simply weight the pros and cons of changing notation:
- The notation is now more in tune with measure theory
- The notation is harder to learn
- The new notation requires mathematitians to "unlearn" the old notation from their subconsciousness (or else it will cause them to "think" more slowly since they will have to translate)
- The new notation confuses engineers and other professions which use mathematics without going to the deep theoretical stuff. It also reinforces the view by these people that mathematics is mental masturbation for its own purpose.
Do you really think using only integral notation will be any better? Let us say I have a sequence $a_n$ and a function $f(x)$. If I want to integrate the function and sum the sequence, I write
Let $a_n$ be a sequence and $f$ a real function. Then we calculate $$\int_a^b f(x)dx + \sum_k^l a_n$$
And it is immediatelly clear what is happening. In your case, I would have to say
Let $a$ be a function from $\mathbb N$ to $\mathbb R$ and $f$ be a function from $\mathbb R$ to $\mathbb R$ and let $\mu$ be the Lebesgue measure on $\mathbb R$ and $\lambda$ the counting measure on $\mathbb R$. Then we calculate
$$\int fd\mu + \int ad\lambda$$
Is the second way really better than the first?