Both 尽管 (jǐnguǎn) and 即使 (jíshǐ) are used to emphasize contrast in Mandarin Chinese, and both can be followed by 却 (què) or 可是 (kěshì). Both of them can be translated as 'even'.
They are used for different situations, though. The difference is that 即使 is used for hypothetical situations, whilst 尽管 is used to connect unexpected facts.
即使: even if
即使 is for hypothetical or conditional statements, and describes unlikely or extreme situations.
Jíshǐ nà jiā fànguǎn zuò de cài hǎochī jí le, wǒ què bù xiǎng zài nàli chīfàn.
Even if that restaurant cook very tasty food, I wouldn't like to eat there.
Jíshǐ yǒu bǐ nǐ piàoliang de rén, yě bùhuì duō.
Even if there are people more beautiful than you, there won't be many of them.
Jíshǐ wǒ yǒu zúgòu duō de qián, wǒ yě bùhuì mǎi yí liàng fǎlālì.
Even if I had enough money, I still wouldn't buy a Ferrari.
尽管: even though
尽管 is used to link statements of fact, i.e. where the situation is already as described. In all of the example sentences above, 即使 could grammatically be replaced with 尽管, which would change the meaning to:
"Even though that restaurant cooks very tasty food, I still don't want to eat there."
"Although there are people more beautiful than you, there aren't many of them."
"Even though I've got enough money, I still won't buy a Ferrari."
More examples for 尽管:
Jǐnguǎn Měiguó shì ge fādá guójiā, què hái yǒu xiē hěn pín qióng de rén.
Even though America is a developed country, it still has some very poor people.
Jǐnguǎn hē jiǔ duì shēntǐ bù hǎo, hěn duō rén háishì xǐhuān hē .
Even though alcohol is bad for health, a lot of people still like to drink it.
Jǐnguǎn tā de kǎoshì chéngjì bùgāo, bìyè hòu háishì hěn chénggōng.
Although his exam scores weren't very high, he was still successful after graduation.
Translated from: eastasiastudent.net