I'm not a native speaker, but my interpretation is this:

• obvious means that something is so easily visible that there is no arguing about: you don't need to search for the reasons, they are in plain sight.
• evident means that many things point to it. Those things might not be visible without searching, but once they are found, it's simple to come to the state conclusion.

The presence of an elephant in a room is usually obvious.

The presence of a mouse in a room is evident, but you might be able to miss it if you don't look close enough.

I'm not a native speaker, but my interpretation is this:

• obvious means that something is so easily visible that there is no arguing about it: you don't need to search for the reasons, they are in plain sight.
• evident means that many things point to it. Those things might not be visible without searching, but once they are found, it's simple to come to the stated conclusion.

The presence of an elephant in a room is usually obvious.

The presence of a mouse in a room is evident, but you might be able to miss it if you don't look close enough.