1)Why do entrepreneurs and VC valuations differ so markedly? 2)What should Pete Perez do? How much in terms of equity percentage is VC sponsorship worth? What would you do as an entrepreneur?

1) Entrepreneurs being the founders of the firm and seeking external finance are wanting to value their firm as high as possible due to the following reasons: -

a. High Valuation means more funding and hence more growth.
b. They are highly optimistic about their venture and often tend to ignore negative/difficult times that comes with businesses.
c. If the valuation of the firm is high, they will have to dilute minimum possible ownership for the same amount.

Venture Capitalists being professional investors tend to consider an overall scenario, they value the firm on a pessimistic or a very strict approach, so that if things go wrong, minimum possible returns can be harvested.
The lower the present value of firm, the higher benefits shall be reaped by investors in future in the times of growth.

NOTE : only question 1 has been answered adhering to chegg guidelines (and to answer question 2, we need more information)