I cannot distinguish and therefore mispronounce /ae/ and /a:/. Is there any different?
It varies with the region and dialect, etc but for my midwest American English, it is the difference between cat and father.
they are very different. Can't you hear the difference between these two? Click on them:

/ae/ => /caet/ --- CAT
/a:/ => /ca:t/ --- COT

You can check this out too: American Vowels
That comes from this webpage http://www.americanaccent.com/pronunciation.html , which talks about some important feature of American English (it's Ann Cook's website). I don't know if you are interested in American English or British English, but I'll give those links anyway... someone might be interested.
Emotion: smile
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The first one appears in cat, for example, whereas the latter one is a long vowel, for example, the one in a car.
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