Putting aside the issue whether 'extremely similar' (please define what you mean by this; do you mean within the sampling error?) is the same as 'coincide', I believe the answer is no. To be equivalent, they must imply each other in both directions. If outputs are independent as in ICA (but not Gaussian, because ICA does not work for (circular) Gaussian data), then they are uncorrelated as in PCA. The converse is not always true. So one can only compare both methods for non-Gaussian data. For these, since both methods operate on different moments, they are not equivalent. ICA gives generally oblique components, while PCA components are always orthogonal. Therefore, orthongality (i.e., E(X'.X)=0) is a necessary condition for equivalence. But E(Xi'.Xj)=0 does not imply E(Xi'^2.Xj)=0, E(Xi'. Xj^2)=0, etc. for all higher-order moments and n-tuples to vanish as in ICA, so it is not a sufficient condition.