Apparently looking at the leaves from certain distance give this:
Is this the problem with the mod or the outdated texture pack, or something else?

With near view distance:

I have a similar issue. You can see when the leaves becomes nicer when the resource pack is activated as it only works wehn you're pretty close.

This is caused by sodium's "Use block face culling" option.
Just disable it and you're probably good to go.

Yep, Motschen was right, disabling "Use Block Face Culling" fixed the problem
Should have look around the settings first