The meanings and explanation of なんだか and なんとなく in Wikitionary:
It seems that they have totally the same meanings. Is there a difference between the usage of these two adverbs?
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They mean very similar things. However, they are not totally interchangeable. The other definitions in the link you gave shows how they can be different.
For なんとなく, there is no particular reason (or the reason is too petty to be considered). For なんだか, the speaker doesn't know the particular reason. This is why, the following sentences are correct:
But you can only use なんとなく in the following sentences:
And why, the following phrase sounds more romantic:
This article summarizes it well, IMO.