Alice was deeply hurt because she wasn't invited.
The grammatical relationships in the sentence above might be a little easier to see. The clause "she wasn't invited" is the argument of the word "because". Traditionally we label "because" as a subordinating conjunction. In turn, this subordinate construction is a general modifier. What it modifies is ambiguous. It might have an adverbial role, modifying the adjective phrase "deeply hurt". It might have a supplemental role, modifying the entire independent clause "Alice was deeply hurt."
Alice was deeply hurt that she wasn't invited.
The word "that" can also serve as a subordinating conjunction. Its use in this sentence is less ambiguous, the subordinate construction taking a strictly adverbial role. It directly modifies the adjective "hurt".
We can see this difference when we change the order of the sentence elements:
Because she wasn't invited, Alice was deeply hurt.
*That she wasn't invited, Alice was deeply hurt.
The subordinate clause marked by "because" works as an introductory element. The one marked by "that" does not. Similarly:
Alice was hurt deeply because she wasn't invited.
?Alice was hurt deeply that she wasn't invited.
Separating the subordinate clause with a "that" from the word that it modifies leaves us with a sentence that does not seem either natural or particularly clear.
In this context, reason or cause (as marked by "because") and manner or nature (as marked by "that") are practically indistinguishable. "She wasn't invited" answers both why she felt hurt and how she felt hurt. The two sentences carry much the same overall meaning.