While you wait
While you wait is an extremely well-established stock phrase for business services. So, echoing the phrase can trump other grammatical pressures in a sentence.
J.R.'s answer here has some nice illustrations.
Seeking the subtle difference
You are right that this sentence would also be grammatically correct:
Customers can now sit down comfortably while they are waiting for their number to be called out.
However, even without the influence from the familiar phrase while you wait, I think the simple-present wait is more appropriate here. Here's another example:
I'll read a book while she goes to the store.
You could also say is going, but I think goes is slightly better. I'm not completely sure, but after sitting here thinking about it for a few minutes, my best guess is that parallel tenses are better for indicating that the actions named by both verbs occupy the same span of time. For example, your original sentence suggests that a customer would remain seated for the full duration of the waiting. The version with while they are waiting dimly suggests that the customers might start waiting, and then sit down while they are still waiting, and then stand up and walk around while they are still waiting, and then maybe sit down again while still waiting, and so on.
"I'll read a book while she goes to the store" suggests that the speaker's reading a book and her going to the store will occupy (approximately) the same time. "I'll read a book while she's going to the store" suggests that she might start going to the store, and then in a few minutes I might start reading while she's still going to the store, and then I might finish reading the book while she's still going to the store. I might be done reading the book ten minutes before she returns from the store.
Even if this is the true explanation (I'm not sure it is), the implication that both actions occupy the same span of time is definitely vague. It's not something to treat as a rule.
(I'm a native American English speaker.)
The present continuous, and not the simple present, is called for in both of these sentences.
We use the present continuous (or present progressive) to talk about events or conditions that are in progress or unfinished at the time of speaking. We use the simple present to talk about events or conditions that take place in general.
When we say "It rains", we say only that rain falls, without any reference to the time of the event. When we say "It is raining", we say that the rain falls as we speak.
Thus, the present continuous is correct in your first sentence:
It is raining, why don't you take your umbrella?
It is sensible to take the umbrella if the rain is in progress at the time of speaking. If the undeniable fact that "it rains" were a plausible reason to carry an umbrella, everyone would carry an umbrella every moment of every day!
In your second sentence, the same principle applies. When we say "You wait", we say only that the subject waits, without any reference to time. When we say "You are waiting" (or, with subject–auxiliary inversion in the interrogative form, "Are you waiting"), we say that the person waits at the moment when we speak.
Thus, the present continuous is also correct in your second sentence:
Who are you waiting for?
Best Answer
The verb wait is a dynamic verb which is also known as action verb or activity verb and it is opposite of stative verb.
One of the biggest differences between the Simple Present tense and Present Progressive (Continuous) for a dynamic verb is the former has a habitual aspect and the latter expresses incomplete action or state in progress at a specific time.
For example, let's contrast "I run" with "I am running". The former has a habitual (repetition) aspect and could be an answer to a question, "what kind of (physical) exercise do you do". "I run" means I run habitually to stay healthy. However, the latter means I am running now which is still in progress but we expect the action to finish soon and it could be an answer to "What are you doing (now)?". There is a huge difference.
Going back to your question, "I wait for your answer" can't express your progressive action you are performing at this moment. Nobody waits for an answer habitually or repeatedly. It happens only from time to time and doesn't happen on a regular basis. Therefore, it is more appropriate to use "I am waiting for your answer".
Present tense also has other aspects and it can mean a future action when you use the verbs like leave, start and depart, etc. For further information, please read the linked Wikipedia article.