The article is blocked behind a paywall. Nonetheless, I think the major terms and components can be addressed based on your description.
Propensity score weighting does not weight by the "odds" or weight by the "inverse". Propensity score weighting weights observations by the inverse of the probability of receipt of the treatment.
A difference-in-differences is an estimand, not a response variable. The advantages of ANCOVA, modeling the outcome adjusting for baseline values as a covariate, over a change-score approach have been discussed several times on this site. See here for a lively and thorough discussion. Even so, the difference between the two approaches is a fixed effect vs. an offset; thus the outcome is always just the response variable; hence the formatting of the response variable and interpretation of the treatment receipt coefficient as a difference-in-differences is the same in both approaches.
The average treatment effect on the treated and the average treatment effect (on the sample) is not a designation I've heard before. By definition we estimate the ATE by subtracting a comparable set of differences that would be found in an untreated group. In a clinical study this would be called Hawthorne effect, in observational studies this is usually a type of prevalent case bias. Together, they are types of pre/post differences that do not arise as a form of confounding, so it is not addressable by propensity score weighting.
Conversely, regardless of the presence of these effects, confounding by indication is capable of exaggerating (or attenuating) treatment effects. Propensity score methods (matching or weighting) are still needed to control for confounding effects.
Best Answer
Despite some similarities, propensity score matching (PSM) and inverse probability of treatment weighting (IPTW) behave differently, mainly because matching selects some cases/controls and discards others, while IPTW includes all study units.
The scholarly literature suggests indeed that PSM and IPTW have similar accuracy in many cases, but in some specific scenarios PSM behaves better. However, in my experience when there are discrepancies between these methods, eventually the data collection approach and the study itself ends up being less credible and externally valid.
In any case, you can peruse the following works on the subject (it is only a quick selection):
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5564952/
https://elischolar.library.yale.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1347&context=ysphtdl
https://www.sciencedirect.com/science/article/pii/S073510971637036X