Difference between a multiple regression and several linear regressions? Context: I'm an undergrad who took Intermediate Econometrics more than a year ago and I'm trying to brush up on some of the skills.

As I was reviewing multiple regression, I realized I didn't quite understand what makes the regressor coefficients different when doing a multiple regression vs doing a univariate regression. For instance, if I were to regress wages on years of education, why would the coefficient on years of education be different than when I regressed wages on years of education and test scores?
The data might be correlated which can lead to simple and multiple linear regression models giving different results. 

Which goes against the assumption of no multi-collinearity:
Multicollinearity occurs when independent variables in a regression model are correlated.

When modeling years of education to wages, the results may show significance but modeling both years of education and test scores could show years of education as not significant because years of education and test scores are correlated. 

Try  plotting years of education and test scores.