According to the standard German dictionary Duden, they are interchangeable in everyday language.
Technically, however, zahlen refers to the amount that you pay, while bezahlen refers to what you are paying for. E.g.:
Ich bezahle das Abendessen = "I am paying for dinner"
Ich zahle 50 Euros = "I am paying 50 Euros"
Here you can see why it's "colloquial" to interchange them. Technically, if you swapped them in the last examples, you would get:
Ich zahle das Abendessen = "I am giving you dinner as a reward"
Ich bezahle 50 Euros = "I am buying 50 Euros from you [i.e. with US$]"
In the context of this sentence, either would be perfectly fine.
Not all Germans yell. In fact, that is very a stereotypical view of Germans, as Jemol2 mentioned. German is often interpreted as an "angry language" because of World War II, during the time where Adolf Hitler was a ruler in Germany and he has been one of the most commonly documented subjects that are from Western and Central Europe. Please do not associate Germans with this stereotype, or associate anyone with any stereotype, because it is often seen as rude and can be quite hurtful to those who speak this language or anyone who has been stereotyped. I thank you if you take my advice. In fact, if you'd like to learn more about why and how this incorrect stereotype is so commonly believed to be true, check out this article. http://sites.psu.edu/mckeerclblog/2013/04/17/why-are-germans-so-angry/
I'm so glad you wrote this comment. As an older American of German descent (mit 62 Jahre), my observation is that we see little if any modern media (TV series, documentaries, movies) translated from modern Germany. There was one recently distributed, "Deutch 82" which covers the time just prior to the fall of the Berlin Wall. Of course, as of now, kids and young adults cross from one side of Berlin to the other with little need or thought of the wall which once divided the city. The same is true of Americans with respect to slavery. The difference is that white Americans in particular are not unfairly shackled to slavery as even modern day Germans are to WWII. It is actually really unfair to Americans to do this to ourselves, since the largest single group of Europeans to immigrate to the US came from Germany. The influence of Germany on American culture, however fragmented today, cannot be extricated. And it shouldn't be. It accounts for much our greatness, let alone the scientists, doctors, engineers and artists whom the US snapped up immediately after the end of WWII, as did the Russians. Nothing will ever justify nor excuse the Holocaust. But I believe it's more than safe to say history has exonerated the German people who in one of the country's lowest points responded very characteristically in a nationalistic, if misled, manner; much as the soldiers of our Viet Nam war followed in their fathers' footsteps. If anyone still believes the average German of today should be harshly judged because of what happened in WWII, then he is either misinformed or decidedly bigoted. Few if any countries in the history of mankind devote so much money to education and research. To this end, bearing in mind WWII wasn't the first time the Jews were persecuted, and more importantly, Russia revealed its true part in WWII as a long-term cold war opponent, and -- again without letting Hitler and the SS off the hook -- plenty of historical evidence supports the notion that we were actually fighting the wrong enemy. Winston Churchill said as much when he wrote "The Americans always get things right, after they get everything else wrong."
I don't understand. You have the translation of "Ich bezahle!" (which is a declarative statement that "I am paying/will be paying [for the purchase at hand]).
You want it to mean "I would like the bill," and you've been given a German translation for this unrelated sentence. I can confirm that "Ich bezahle" does not correspond to this.
If you're looking for slang, you can tell the waiter "Quittung, bitte", which means "check, please."
Be careful not to map related meanings onto simple phrases. It's easy to do in the beginning but it can slow you down later when you want to add nuance but believe it exists in simple phrases.
Ich bezahle means specifically handing over the money or check in exchange for something--the paying.
While "I'm buying!" can idiomatically mean that you're paying for something in specific circumstances (like when you're at a restaurant and offering to pay for a round of drinks for your friends), they're not identical. For example, you probably wouldn't walk into your utility company office and pay for your electrical bill and say "I'm buying!" And you wouldn't walk into a grocery store and say "I'm paying for milk."
So it's important to know that bezahlen means "to pay."
And that's different from kaufen which means "to buy."
So it's better to learn the standard definition of bezahlen and then understand why it's used in an idiom. (Or that it's used! You never can tell with some idioms!)
There is no "am" in German. Using the "bin" as "am" is WRONG. Both "I pay" and "I am paying" are translated exactly the same in German ("ich bezahle") and you cannot tell which is which without context.
No continuous aspect
In German, there's no continuous aspect, i.e. there are no separate forms for "I drink" and "I am drinking". There's only one form: Ich trinke.
There's no such thing as Ich bin trinke or Ich bin trinken!
When translating into English, how can I tell whether to use the simple (I drink) or the continuous form (I am drinking)?
Unless the context suggests otherwise, either form should be accepted.
Despite the fact that this probably doesn't make sense in German (and I usually hear it the other way around in English: "you fly; I buy"), there's one very important thing you've missed! Fliege means "I fly." (bezahlen doesn't mean buy, but it does convey the meaning in the English expression: if the other person does the work of driving or picking up (usually take-out food), then the speaker will pay, so it's a better choice than kaufen)
So it should be: ich bezahle, du fliegst.