The "score" at least seems doable if this is a well-studied differentiation. If you can define initial, terminal, and intermediate states from either a single-cell study, or a timecourse study that examined this particular differentiation, then you can do something like:

1. Compute a small number (<=100) of landmarks between the reference states
2. Summarize landmark expression into an expression vector
3. Compute the correlation of your observed expression to each of the landmarks (ideally this should be positive)

These relative correlations would basically be a "score" along the differentiation path; though it only really makes sense if there's not some huge batch effect that renders all of the correlations small or negative.

However, I'm not sure what you mean by the effect of a driver gene -- unless you are specifically over-expressing a single gene, I don't think there's a way to apportion any score among different genes; you can only measure total effect.