From the Normalizer docs:
Each sample (i.e. each row of the data matrix) with at least one non zero component is rescaled independently of other samples so that its norm (l1 or l2) equals one.
And StandardScaler
Standardize features by removing the mean and scaling to unit variance
In other words Normalizer acts row-wise and StandardScaler column-wise. Normalizer does not remove the mean and scale by deviation but scales the whole row to unit norm.