Cycloaddition vs Electrocyclic Pathway
The Woodward-Hoffmann rules only tell you whether a reaction is thermally allowed or not, they do not give any indication of the relative rates of multiple allowed reactions.

I guess you're asking why the 4π component in a [4+2]-cycloaddition doesnt undergo a conrotatory 4π electrocyclic ring-closure? This is plausible, but I suppose even if it did happen the cyclobutene product is highly strained so the equilibrium would lie in favour of the ring-opened diene anyway.