How can I compare two recurrence relation to find which one is better on large inputs
The simplest way is to solve both recurrence relations, and see whether one of them grows faster. In your case there is actually no need to do that, since we can observe that on the one hand
$$
T_1(n) \geq 2T_1(n-1) \geq 2^2T_1(n-2) \geq \cdots \geq 2^{n-1}T(1) = 2^{n-1},
$$
and on the other hand
$$
T_2(n) = O(n^2),
$$
by the master theorem. This is enough to deduce that $T_2(n) = o(T_1(n))$, that is, $T_1(n)$ grows faster than $T_2(n)$.