You know, that's an interesting comparison. As much as Q exasperated Janeway she would sit down with him and have actual conversations. I can't recall that ever happening with Picard.
As much as folks talk about Janeway being stubborn and only wanting things her way, she took the time to learn things from Q because she did take time to have conversations with him. Picard acted like he knew more than Q and never took the time to think: what can this Godlike being teach me? That is what Q was trying to do, teach Picard the whole time. For an Ambassidor, Picard wasn't diplomatic at all when it came to Q. Janeway was.
That probably had more to do with Q being more lethal in his interactions with Picard than he ever was with Janeway.
Lets review...during his first encounter with Q, his helmsman was frozen and nearly killed. Q then kidnapped his bridge crew, put guns to their head and put them on trial for human history.
In a subsequent encounter, Q once again puts his brdige crew in danger by exposing them to some Napoleanic animal things (who proceed to impale Wesley and hurt other crew members) while simultaneously delaying the Enterprise from a life or death mercy mission which ultimately resulted in deaths amongst the colonists.
The next encounter results in Q flinging the Enterprise to the Delta Quadrant. Exposing the Enterprise and the Federation to the Borg and resulting in the DEATH of 24 crew members.
Why exactly would Picard hold Q in anything less than total disdain at that point. Q was not just a nuissance, he was deadly. Hell, the events of "Q Who" directly lead to Picard's torture ar the hands of Borg and the massacre at Wolf 359.