Is there a difference between 担心 (dānxīn) and 着急 (zháojí)?
There are some differences between these two words. 

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  “着急” 
  Used as an adjective:
  1. Something emergent happens (in other words, something horrible or fatal is very likely going to happen), and you feel upset. For example, when you lost your kids or you're going to be late for your work.
  Example 一位母亲因为找不到她的孩子而非常着急。(Can't use "担心")
              A woman is very upset because she can't find her kid anywhere.         
  
  2. While doing something, you only care if you can finish the work as soon as possible and may forget something you should notice, thus causing the work to be done badly.
  Example 老师嘱咐我们考试做题的时候不要太着急。(Can't use “担心”） The teacher told us not to be too hurry when when answering the questions.
  used as verb or adv: means the same when used as adj. 
  
  Used as a verb:
  Use it as a verb only when nothing emergent happens, yet in the future things are going to be pretty bad very likely, and you're sure that it's deadly. You're hopeless or you know doing something though won't likely help you, but still possible to help you out, therefore you'll try your best to avoid it.
  Example 儿子因父亲日益加重的病情而着急（Can't use “担心”）。 
  The son feels upset for his father's deteriorating illness.
  
  “担心”
  Used as a verb: You're worrying that something bad might happen in the future, but you comfort yourself, reasoning that it possibly won't happen. Or you know it's very likely to happen, but you also know that it's not a big deal since it will get fixed if it happens. For instance, if you're worrying about your kid's future, you can use this word to describe yourself. 
  Example 1 他车开的太快了，我担心他会出事故。（Can't use 着急. it's likely to happen, but not that likely. It's a big deal once it happens though. And the other reason is that 着急 is a intransitive verb.）
  He drove the car too fast, and I was worrying that he would have an accident. 
  Example 2 儿子马上就要出国留学了，父亲为他在外的生活而担心（You can use 着急 too, and actually it's better than 担心 in this case since it shows that the father really cared about the son.)
  The son is going to be studying aboard, and the father is worrying about his daily life.
  孩子出门上学的时候雨下得很大，他的父母为他的安全而担心（interchangeable with 着急 since something bad would very likely happen to the kid)。
  It rained fiercely when the kid went out of the home for school, and the parents were worrying about the kid's safety. 
  
  Used as an adj: the same when used as a verb. 


Summary: 
1. “着急” is more intense than “担心”. When something terrible or may cause death or injury will happen very soon, you can only use “着急”. 

2.Under some situation, they're somewhat interchangeable. For example, when you're worrying about someone's health, security, etc., as long as a catastrophe is not sure to happen, or it is very likely to happen but not that serious, you can use both words. But here "着急" is more intense than "担心", because "着急" shows that you really care about someone, or you really don't want that bad thing to ever happen. And If you use “担心”, it reveals that though being bothered, you're comforting yourself that it probably won't happen. 

3. “着急” also means that you're eager to finish something, which is not one of the meanings of the word "担心".

4. 着急 is an intransitive verb, and 担心 is a transitive or intransitive verb.

5.着急 is often used as an adjective while 担心 is often used as a verb.

6.To answer your question concerning the variation between north and south: These two words are both used in the north quite frequently. As to the southerners, I think they should use these words often too, as these two are basic words in Mandarin. Some place in the south may use some uncommon words for each of them. 

7. Certainly 着急 is informal, but 担心 is considered as formal, and it often appears in newspapers and broadcasting. 
So remember in formal situations, use 担心 when both of them make sense. When you want to express the feeling that could be described by 着急 when writing a news report, use 焦虑，焦急，紧张。

I'm not an expert on this, so some contents may be inaccurate. Any edition or correction is welcomed.

Synonyms for "担心": 忧虑，着急，忧愁，担忧  synonyms for "着急"：焦虑， 焦急，匆忙，紧张