In Japan, anytime you hand someone cash in personal life it's rude not to put it in an envelope. Examples includes gifts or paying your Japanese teacher.
The tray avoids passing cash hand to hand. 20 years ago, clerks (or taxi drivers) would be visibly surprised and slightly shocked if you tried to hand them cash. But that's relaxed a lot now. Tourists hand cash to clerks and the reaction is slight if at all. Taxi drivers often exchange cash hand to hand. It's still more polite to use the tray but less strict.
The tray avoids passing cash hand to hand. 20 years ago, clerks (or taxi drivers) would be visibly surprised and slightly shocked if you tried to hand them cash. But that's relaxed a lot now. Tourists hand cash to clerks and the reaction is slight if at all. Taxi drivers often exchange cash hand to hand. It's still more polite to use the tray but less strict.