The disk is a hard copy of the original (or in the case of TTK, current) content, which is simply used to distribute the content to players who prefer none digital purchases.
As for the digital purchase same thing your buying access to the content. Then to play said content you agree to the EULA. Then when down the road another dlc comes out you repeat the same process.
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A dlc isn't a new game. The old content should still be available
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And its still avaible they even added new stuff for you (revamp strikes) you are locked out of the content you didnt pay for. You can still play year 1 strikes, poe (if you have HoW), ToO (when avaible), crucible, VoG, CE (if you have TDB), IB