I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is adapted but the converse is not necessarily true) but I can't get it from the mathematical definitions.

The definition of *progressively measurable* process is the following:
A stochastic process $X$ defined on a filtered probability space $(\Omega ,{\mathcal F},{({{\mathcal F}_t})_{t \ge 0}},P)$ is progressively measurable with respect to ${({{\mathcal F}_t})_{t \ge 0}}$, if the function $X(s,\omega):[0,t]\times \Omega \rightarrow \mathbb{R}$ is $\cal{B}([0,t]) \times \cal{F}_t$ measurable for every $t\ge 0$.

The definition of *adapted process*:
A stochastic process $X$ on $(\Omega ,{{\mathcal F}},{({{\mathcal F}_t})_{t \ge 0}},P)$ is adapted to the filtration ${({{\mathcal F}_t})_{t \ge 0}}$ (or ${\mathcal F}_t$-adapted) if $X(t)\in {\mathcal F}_t$ for each $t \ge 0$.

Can someone explain me the difference in simple words? I think I understand the definition of the adapted process quite well but I'm probably confused of the role of the Borel sigma algebra in the definition of the progressively measurable process.