Basic Weaknesses form the pool of random weaknesses you draw from when building your deck. Thus, any investigator can end up with any Basic Weakness.
All other weaknesses can only be added to your deck when you're explicitly required to do so (usually by your investigator's deckbuilding rules).
To be precise, you also only add Basic Weakness to your deck when told to do so. It's just that every investigator is told to do it as part of their deckbuilding.
If an effect says "basic weakness" then it only means one of those 10 cards that say basic weakness.
If an effect says "weakness" then it refers to any of the 15 weaknesses (basic or not) in the core set.