Will functionality for the Original Xbox games via Xbox One Backward Compatibility change from the Xbox 360 games in any way?

Functionality will be very similar. You can play the digital or disc-based game you own, taking advantage of Xbox One features like Game DVR and broadcasting.

Since these are the original games, not remasters, Xbox Live services such as online multiplayer and in-game marketplaces that were available on the Original Xbox, will not be available. However, offline multiplayer scenarios such as co-op, party scenarios (multiple controller on one console), and system-link are supported if they were supported on the Original Xbox. With system-link (if game supports the feature), you can play with others across Original Xbox, Xbox 360, Xbox One, Xbox One S and Xbox One X consoles.

There are no Xbox achievements for Original Xbox games because this functionality was not available on the Original Xbox at that time of its release. Lastly, game saves will not transfer to Xbox One but once created on Xbox One, the game saves can transfer to other Xbox One consoles through the cloud.