When should you use 尽管 (jǐnguǎn) instead of 虽然 (suīrán)?
Let me translate an answer I saw on a Chinese website.


  这两个关联词语的使用是有差别的.词典中的注释很明了,"尽管"表达的是让步转折,即说话者已经认识到一种客观存在,这种客观存在应导致某一种结果；但他依然坚持自己认为正确的另一种结果。可谓坚持己见，态度坚定，大有死了的鸭子嘴硬不肯悔改之意。“虽然”是客观陈述，“尽管”是主观表达。
  如：虽然天冷路滑，学员们还是按时到达了；尽管天冷路滑，你们也要按时到达。


尽管 and 虽然 are somewhat different. "尽管" implies personal opinion (similar to "although"?), indicating that the speaker understands that the situation usually leads to result A, but insists on result B nevertheless. "虽然", however, does not carry this connotation because it goes with 但是 and can be treated as "but". "虽然" merely states a fact.

For example,


  虽然天冷路滑，学员们还是按时到达了。
  It was cold outside, but the students arrived on time.
  
  尽管天冷路滑，你们也要按时到达。
  Although it is cold outside, you should arrive on time.



As a native speaker, I think 虽然 and 尽管 are interchangeable in the first example, while 尽管 is the best way to put it in the second sentence. Thus, when you are expressing strong personal opinion, 尽管 is a better choice.
尽管 is also more formal. People hardly use 尽管 in colloquial language. (Probably we omit 虽然 as well, as 但是 will do in most situations.)
尽管 has an additional meaning, which Terry explained quite clearly in his post.