We adjust for observable characteristics when comparing outcomes across groups because those characteristics affect outcomes and differ across groups. A linear regression makes a strong assumption about the relationship between a set of explanatory factors X and an outcome y, like y = Xb+e. When that equation is not quite right, all the estimates can be wrong. One way to sidestep a strong assumption about the functional form of the relationship is to match on all variables except the variable that defines groups, call it treatment x. It turns out that matching on the conditional probability of treatment x, called the propensity score, is just as good as matching on all variables, and far easier.
However, matching on the propensity score is equivalent to forming new weights where each match gets weight one for each time it is chosen as a match, and it turns out that other weighting schemes are even better than simple matching. These propensity score reweighting schemes are similar to methods used to adjust survey weights for nonresponse.
For propensity score matching and reweighting methods to work, we need the conditional probability of treatment x, the propensity score, to be bounded away from 0 and 1 (we can’t compare a treated case with conditional probability of treatment x of 1 to any untreated case because there can’t be any, and likewise for probability 0 cases). We also need the two groups to have propensity scores over the same range, an assumption called overlap, so there are comparison cases in the untreated group for each treated case, and comparison cases in the treated group for each untreated case.
It is important to remember that the assumptions about selection bias are the same in both linear regression and propensity score matching and reweighting methods, namely that any important selection into treatment depends only on observable characteristics, not factors we do not observe. It is possible to reduce bias using linear regression or propensity score matching and reweighting methods even when selection into treatment depends on factors we do not observe, but it is also possible to exacerbate existing bias.