What's the difference between those two? For instance, in a sentence like "你们赶紧/快走吧" can I use both indiscriminately? What are the particular conditions that make me prefer one rather than the other?
赶紧 is an adverb and modifies the verb that follows it. It has the meaning of "hurry up to do something".
赶紧走 hurry up and go
赶紧吃 hurry up and eat
赶紧跟她道歉 hurry up and apologise to her
快 in you example sentence functions as an adverb. It has the meaning of "do something quickly."
你们赶紧走吧 "Hurry up and go" and 你们快走吧 "go quickly" means essentially the same thing.
But the word 快, in other contexts, can also be an adjective, which you can modify using 很，太，非常, etc.
But you cannot do the same with 赶紧. This usage of 快 is very different from 赶紧.
In the usage of the sentence
Yes you can use both indiscriminately.
The only different between “赶紧” and “快” in the sentence above is the number of words.
赶紧means you have to do it now
快got other usage but one of them is same as
In my opinion, if I have to use
快 in the above sentence it would be like I'm shouting
快走吧！！ shorten the number of sounds I have to pronounce.
Then I will use
赶紧 in the above sentence it would be like I'm talking gently
There are excerpts from The Contemporary Chinese Dictionary,which is authority of the Chinese language.
First，We talk about the discussion of answer above.
Though 'fast' and 'past' can be translated respectively as 快的 and 过去的, in Chinese, no expression like 时间过的真快. As the dictionary says in Interpretation 3: 的 used after a predicate verb to emphasize the perpetrator,time,place or method and the usage is only legal in past tense.
For 得，in interpretation 3: used after verb or adjective to express the result or extent. If we directly translate the sentence into English:
时间 过 得 真 快 Time passed really fast
得 has no corresponding translation.
Noticeably, both of them is light tone in the circumstance.
In addition , in some ROC period or earlier literatures we can see they used 的 instead of 得.
Okay,let's talk about 赶紧 and 赶快. They both are adverb, the prior means seize the opportunity firmly, without any delay and the later means to seize the opportunity,speed up.
- In an emergency, we use 赶紧 more, such as 赶紧走！(Go quickly!),赶紧加工（Accelerate the process of machining）.
- In some situations we also use 赶快: 赶快，否则你会迟到！（Hurry up, or you'll be late!），赶快上车（Get on quickly）。
A bit confusing, actually, there are no so many differences between them. Often one can be replaced by the other.