If Legendre's conjecture is true, couldn't we easily obtain $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$ where $p_{n}$ is the $n$th prime?
$$p_{n+1}<(\lfloor \sqrt{p_{n}} \rfloor + 1)^{2}<( \sqrt{p_{n}}+ 1)^{2}$$ $$\sqrt{p_{n+1}}<1+\sqrt{p_{n}}$$ $$\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$$
which is Andrica's conjecture.
Legendre's conjecture is reported by Wikipedia to be that for each $n$ there is a prime between $n^2$ and $(n+1)^2$. That would imply $$ \lfloor \sqrt{p_{n+1}} \rfloor - \lfloor \sqrt{p_n} \rfloor \le 1. $$ For example, $\lfloor\sqrt{173}\rfloor=13$ and $\lfloor\sqrt{167}\rfloor = 12$. Obviously in many cases the difference between integer parts of the square roots of consecutive primes is $0$.
Notice, however, that $\sqrt{1021}\approx31.953\ldots$ and $\sqrt{953}\approx30.87\ldots$. Nothing in Legendre's conjecture says these cannot be consecutive primes, but Andrica's conjecture rules that out.