Assess/ Weigh

Not really. 'Assess' may seem more formal.

Suppose a business proposal is being discussed:

"He is assessing the merit of the proposal."
"He is weighing the merit of the proposal."

Could it still be that the only difference is the formal-ness of "assess"?

The word you want is 'formality'.

Yes, if even that. However, 'weigh the merits of' is also a collocation.

Thanks for your time and patience, Mr Micawber!

Let me take one more shot at this:

"The jury assessed the evidence before announcing the verdict."
"The jury weighed the evidence before announcing the verdict."

I don't know what to make of this.

They are two synonymous statements. 'Weigh' collocates more highly with 'evidence' than 'assess' does.

"The employer weighed all the given information while assessing each job application."
"The employer assessed all the given information while weighing each job application."

Could one be wrong?

Could one be wrong?-- No.

Thanks Mr Micawber.