They are just different animals when created on the web interface or in the app.
The thing that is bust there is that THS does not make that clear. They seem to be doing their best to confuse members, by using the same term for both, by putting them in the same list.
I prefer the Saved searches that are created in the web interface (on a map), and that generate a daily email. I don’t like the push notifications: often for sits that are already gone, or for sits that are too far away from the location that I was searching for.