Mechanically, what is the difference between a Planeswalker Emblem and an effect that lasts "for the rest of the game" but with no marker or counter indicating that effect (such as on Finale of Revelation, Praetor's Counsel, Sea Gate Restoration, and Stigma Lasher)?

Mechanically, not much. Thematically, a bit.

You seem to be answering a lot of other things, so this might be the ideal time to ask. Why is R&D set on only planeswalkers making emblems when the first planeswalker to produce an emblem, Elspeth Knight-Errant, was worded the same as other cards people are trying to make have emblems, such as Stigma Lasher and Praetor's Counsel? As far I can tell it is non-functional errata sense they do work the same way though i may be wrong...

We emblems to be a planeswalker thing. It is more flavor than function.

There's a discussion about using emblems for things like Stigma Lasher over at Matt's blog, and it struck me as a classic Melvin vs. Vorthos discussion. Melvin wants to use the emblem technology consistently on "for the rest of the game" effects, while Vorthos wants emblems to retain their planeswalker-centric flavor.

I think Vorthos is going to win this one. 

There are a few cards that have effects for the rest of the game, like Stigma Lasher and Praetor's Counsel. Now that you have emblem technology, is this design space that can be explored further on non-planeswalkers?

For now we’ve decided to keep emblems a planeswalker thing.