Hi. Thanks for your question.
I don't know the details behind exactly what Google Maps does, but I can at least try to explain what Esri's tools are doing and some areas where they might be different.
First, it's been a while since I've looked at WMATA's GTFS data, but I recall that in the past it didn't include a calendar.txt file and used a calendar_dates.txt file for everything. This means it is not possible to do an analysis for a generic Wednesday. Instead, you have to pick a specific Wednesday when setting your analysis settings. Please be sure you've checked for this scenario. If you're running your analysis for a generic Wednesday or a specific date that is outside the date range the input GTFS data is valid for, the transit may not be getting used at all, and this would account for the discrepancy. Basically all routes are walking only.
Also, if the routes or Service Areas appear unreasonable, it's possible something isn't configured right in your network dataset. If this is the case, we can try to debug this, or if you care share your data, I can take a look. We want to rule out any network configuration problems before examining any differences in results between Esri and Google.
One difference between Esri's transit solves and Google Maps is that Esri's solvers use the exact time of day configured by the analyst as the starting or ending time. It's hypersensitive to this start time, and the result might be different if you choose a start time a minute or two off. I don't really know what Google does, but since they are designed more for passenger-facing routing apps, I think they're employing some sort of fuzzy start time logic to find the best or most sensible route. If leaving one minute earlier allows you to arrive at your destination 20 minutes earlier (because you don't miss a useful bus), I think Google will find that, but Esri's solvers don't have this functionality. The downloadable Transit Network Analysis Tools may help you to account for this in an analysis.
The difference in street data is also a consideration, although the Streetmap Premium Custom Roads data should be of a high quality and include various pedestrian pathways and useful attributes. Google may have slightly different information about pedestrian paths, although probably not substantial differences that would account for huge discrepancies in travel time.
Hope this helps a little.