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.
The present continuous emphasizes action in real time.
For example, an announcement is made repeatedly over the train-station loud speaker:
The [garbled] will be approximately [garbled] minutes late due to [garbled].
The [garbled] will be approximately [garbled] minutes late due to [garbled].
A passenger waiting in the station could turn to another and ask:
What is he saying?
P.S.
The implication is that speaker and listener are in a situation where this announcement might be made yet another time. The present continuous refers to it as something which is ongoing or repeating.
If the speaker believed that the announcement is not likely to be made again, or is perceiving the announcement as something that took place a moment ago, the question would probably be:
What did he say?
In this particular contextual example, the question would not be asked in the simple present:
unidiomatic What does he say?
unless it referred to a specific place in the announcement:
What does he say after "due to"?
What is he saying after "due to"?
The answer could be, respectively:
He says "due to switching problems".
He is saying "due to switching problems".
The simple present there would refer to the phrase that follows "due to" in the announcement as one that occurs with some predictability or regularity: he says it in the same way each time he says it.
The continuous would refer to the phrase that follows "due to" in the announcement as one that we are hearing him say again and again.
P.P.S.
Even if the announcement comes over the loud-speaker once:
[garbled] [garbled] [garbled] [garbled] emergency [garbled] [garbled] [garbled]
a person might turn to a fellow passenger, even as the announcement is continuing, and ask:
Can you understand what he is saying?
The announcement is in progress.
Best Answer
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:
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:
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.)