Contrast a strong vs weak solution of a stochastic differential equation

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Question

I think I am getting confused by the very similar wording on Wikipedia: “A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space.”

Let’s say we have a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic differential equation $dX_t = \langle\text{Stochastic Differential Expression}\rangle$, with some integral equation $X_{t+s} - X_{s} = \langle\text{Stochastic Integral Expression}\rangle$ where $t \geq 0$, and $s \geq 0$. What precisely are we saying about a solution when it is ‘strong’ vs ‘weak’?