Difference between subgradient SVM and kernel SVM?
You are gravely misunderstanding SVM. Sub-gradient descent algorithm for SVM is a method to solve the underlying optimization problem of SVM.

SVM is always a linear classifier, which can yet by using kernels operate in a higher dimensional space. Therefore in input space, the separating hyperplane (linear!) computed in feature space (kernel!), seems non-linear. 

Effectively you are thereby solving a non-linear classification task, but you are projecting into a higher dimensional feature space where the classification task is solved by a linear classifier. 

Please read: 

Wikipedia on svm

Please watch:

great lecture about SVM