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 will maybe be slightly more employed 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) / ...
