What kind of problems I should resolve with SOM and which with Hopfield?
I've worked with SOMs, but not with Hopfield networks. A quick glance at Wikipedia says that Hopfield networks are used for associative memory, while SOMs are used for dimensionality reduction.
So SOMs are used to visualize high-dimensional data and for clustering high-dimensional data, if the data actually lies on a low (2D usually) manifold in a high-dimensional space. SOMs incorporate a (usually) 2D neighbor-defining grid which is how it does what it does, and at least according to Wikipedia this feature is unique among ANNs.
I guess Hopfield networks could be used for some kind of classification, but their main use would be something more like a robust, approximate value-value kind of lookup. Someone who knows them better could give a better answer.
EDIT: To correct the points addressed by alto's comment.