Has euclidean distance any advantage in compare to another distance based methods like Manhatan distance or Maximum difference metric?
That depends a lot on your use-case.
- If you're working in a continuous space where all dimensions are properly scaled and relevant, then Euclidean is going to be a great choice for distance function.
- If you have a space filled with exclusively countable dimensions, then Manhattan distance will make sense.
Unfortunately, there are rarely clear-cut cases. The best approach is to experiment with a variety of distance functions and see what works best for your data.