Or, is it possible to have two distinct permutations of the first $k$ primes for some $k$ such that using each prime once as a base and once as an exponent to form $k$ terms will yield identical sums?
e.g. $2^2+3^3+5^5+7^7$ is a valid candidate permutation for $k=4$, as is $7^2+3^3+2^5+5^7$; the requirement is that you use every prime in range exactly once as a base and once as an exponent. (Note that you can view this as permuting the bases only while the exponents are in fixed order, or the exponents only while the bases are fixed, or permute both; the effect is the same so long as you follow that requirement.)
There are definitely solutions for less stringent requirements, e.g. if the bases are prime and the exponents are the first $k$ positive integers instead of first $k$ primes, we have
$$3^1+5^2+2^3=5^1+2^2+3^3=36.$$
As per discussion below, I believe this can be effectively restated as:
Are there any two subsets $p,q$ of the primes such that
$$p_1^{q_1}+p_2^{q_2}+\ldots+p_{i}^{q_i}=p_{i+1}^{q_{i+1}}+\ldots+p_k^{q_k}?$$
Any primes in any order may be used, so long as all $p$ values are different, and all $q$ values are different.
Clarification:
There need to be an equal number of terms on each side of that equation.
Otherwise, $2^7+17^3=71^2$ would fit the bill. See also Fermat-Catalan conjecture.
It also seems plausible that requiring the exponents be $3$ or higher would be sufficient.