How to compare performance of Cosine Similarity and Manhatten Distance?
Comparing distance values of different distances is nonsense.

Consider this distance function:

$$d(x,y)=0$$

Clearly, this gives smaller values, but it also is useless.

Consider this trivial variation of Manhattan distance:

$$d(x,y) = \sum_i |x_i-y_i|/100$$

Clearly, this distance is equivalent to Manhattan, but will yield much smaller values.

Choose the distance because it is "the right thing to use", not because of some number. Cosine is good if you desire length normalization and have sparse vectors.