You are implying you have a paired analysis, but you are asking a very general question.

If the data are paired, there's no reason not to use a paired t-test. Furthermore, if you need to adjust for two or more variables, or if order matters in the pairing, you can use regression or regression through the origin. An intercept term adjusts for drift and implies the pairing has a distinct order like pre versus post. Unordered pairs would be found in twin studies. More generally, all these are examples of a conditional likelihood based approach. A "conditional" likelihood simplifies inference by taking advantage of the correlation structure, and use an arithmetic transformation of the response like ratios, averages, or differences to "explain away" the correlation and arrive at independent observations.

A disadvantage of a conditional likelihood approach is that the analysis is restricted to complete pairs. If data are unavailable for one slot, the paired observation must be dropped. Also, depending on the context, it may be a little unreliastic to express a prediction of a difference in a response without explaining what the predictions of the actual response are - such as the case when a drug drops blood pressure by 10 points over 3 months but doesn't report the fraction of patients with SBP < 150 mmHg.

A model for the untransformed response can be used to predict differences (as estimates and 95% CIs), and can use correlation structures like GLS to account for pairings. In a pre-post analysis, it's very common to fit a type of ANCOVA where the "pre" response is actually included as a covariate in the model since it is often treated a baseline condition. This model has the advantage that the "pre" response can be multiplied by the exposure to test for a growth component, since oftentimes the biggest changes occur in cases with the most extreme baseline values such as with weightloss drugs.