What are the differences in the computational power of the deterministic stack and a non-deterministic stack?

Yes, there is a difference between the two. And she's brutal. Non-deterministic stack automata can solve many more context-free languages than deterministic ones.

  I would like to point out that this goes against other automata, where the non-deterministic Turing Machine does not provide computational power beyond that which deterministic manages to provide, just as in the Nondeterministic Finite States Machine can be transformed into deterministic without loss of computing power.

For example, language composed by every parenthesis needs to be closed in order of opening is deterministic. For example, using only curved and square brackets, the grammar would be:

S --> '(' S ')'
S --> '[' S ']'
S --> '(' ')'
S --> '[' ']'

Now, for palindromes, this does not happen. Take a simple palindrome, consisting only of a and b :

S --> 'a' S 'a'
S --> 'b' S 'b'
S --> 'a' 'a'
S --> 'b' 'b'
S --> 'a'
S --> 'b'

How can the automaton distinguish whether it should consider a a terminal as the beginning of a pair creation or the end? For example, the following words are ambiguous:

aabaa
aabababaa

It is impossible for a deterministic stack automaton to determine whether a after b must be to start a new pair or if it matches with a previous.