One conjecture stronger than another?

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So, I was reading the Wikipedia page for Legendre's Conjecture and I came across this:
"Two stronger conjectures, Andrica's conjecture and Oppermann's conjecture"
My question is this: What does it mean for one conjecture to be 'stronger' than another?
 
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Usually it means that if the stronger conjecture is true then the weaker one is true as well. But they are not equivalent.
 

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