I know that both terms roughly translate to 'fashionable' in English. But like there is a difference between 'vivre and habiter', is there any difference between these two terms? Also, 'à la mode' has been adopted into English and refers to desserts served with ice cream. Does the French meaning have anything to do with this?
They have the same meaning, yes, as you seemed to know already.
If any, differences between the two will be minimal in general use. À la mode is a bit less sophisticated and will maybe be slightly more employed in casual speech or when actually refering to (clothes) fashion, but both variants are equivalent for other contexts.
For your second question, no, à la mode isn't used in this sense for ice cream on desserts. I think it's another case of the belgian waffle (unknown as such in Belgium) / crème anglaise (unknown as such in England) / french fries (unknown as such in France) / ...
Not relevant with your question but you can also use "être branché" in a daily speaking conversation.