Turning Machine is far better than a PDA and this can be seen from the following reasons;
Reason No. 1: A PDA is only able to approach the top of its stack, while a Turning Machine can reach any place on an infinite tape. The infinite tape can never be simulated with a single stack, so a PDA is significantly less computationally effective
Reason No. 2: There are algorithms that can be developed with a Turning Machine that cannot be programmed with a PDA.
Reason No. 3: An automaton with use of two stacks instead of just one can simulate a Turning Machine and thus has equal computational power.
Reason No. 4: Turing machines can clarify Context Sensitive Languages while this is not available in PDA.
Reason No. 5: Anything that be done with random-access, parallel, multi-dimensional storage or structured or associative storage can be achieved with a turning machine. All these qualities are not present in a PDA.