Based on BirdNET's recent update, my estimation is that this space is changing rapidly, so take my limited understanding with a grain of salt.
Personally, I prefer the Merlin app for exploratory purposes like teaching myself species ID from my phone in the field in North America. However, when it comes to large-scale audio processing of terabytes of audio on my computer, BirdNET is the best off-the-shelf tool I'm aware of. I think it's important to distinguish between "BirdNET the phone app" and "BirdNET, the platform for scientific audio data processing" (see BirdNET-Analyzer github), the latter of which you can install on your computer and use for scientific research.
The Merlin app does real-time ID -- you don't have to select the portion of the spectrogram you want it to identify (which is currently the case for BirdNET). Merlin doesn't require cellular data once you've downloaded it (BirdNET does; it submits the selected portion of your recording to their servers for ID). I think it's much easier to access my audio recordings from Merlin (whereas accessing recordings from the BirdNET phone app is still more of a process). For Merlin, I have found this eBird help center article useful. That link also has a "Media Best Practices" section that explains how users can upload their Merlin recordings to eBird (I don't believe it is currently possible to upload them directly from the app, but I could be wrong).
This Golden Gate Audubon Society article from July 2021 reviews a few of the differences between the Merlin and BirdNET phone apps, but some of this is already out of date given that BirdNET recently went through a large update that expanded it's ID capacity from 984 North American species to ~3000 species worldwide.
From the article:
Holger Klinck, one of the researchers working on BirdNET, says,
“Fundamentally, both apps are using similar deep neural network
approaches. One difference is that the Merlin’s [Sound ID] algorithm
is exclusively trained with data hosted by the Macaulay Library.
BirdNET uses additional (non-Cornell) data sources.”
BirdNET's algorithm is described in Kahl et al. 2021, whereas I'm not able to easily locate information on Merlin's underlying process (Edited to Add: there are some interesting insights from the Merlin developers buried in Twitter threads, for example here, here, here... perhaps a paper describing Merlin is on the way?). My understanding of the BirdNET phone app, based on the app tutorial section, is that each selected portion of a recording gets sent to the BirdNET servers in Chemnitz, Germany in order to analyze the recordings and identify species (there is an option to "claim authorship" for your recordings). The BirdNET App also has a "submit feedback" button that allows a user to verify whether a species identification is correct or not, which presumably could also be used to update their models in the future.
A recent journal article from some of the BirdNET team (Wood et al. 2022) implies forthcoming updates to future versions of the BirdNET app, which hopefully will enable it to be a more user-friendly community science tool:
"A forthcoming app user data analysis portal will allow users to analyze their own observations in greater detail than the current species list and “explore your area” features, an improvement that will enable users to pursue their own research questions... An optional “point count mode” with which users could submit 3- to 5-minute continuous soundscape recordings would allow communities to document an estimated 5% to 15% of the local bird species per week even with minimal participation 10. Moreover, these acoustic point counts would yield analytically valuable nondetections of the remaining species.
And later on:
"Raw observations are not yet publicly available due to the substantial challenges to hosting tens of millions of observations and audio files; sharing observations only (i.e., prediction scores and metadata but no audio) is simpler but prevents validation of observations and thus leaves the data vulnerable to misinterpretation. Detailed data usage guidelines that will facilitate widespread open-access data sharing are forthcoming."
EDITED TO ADD: more differences between Merlin SoundID and BirdNET are discussed here in this talk by one of the SoundID developers at 57:24: https://www.youtube.com/watch?v=mVhe-MiVwLE&t=3444s
Re your question: Is there a directory of supported species for either?