It's been a forgone conclusion that PS Classics would be making their way to PS4. I continue to be surprised it hasn't happened already. While it's something I'm really looking forward to personally it doesn't really seem like something that would influence the mass market at all. It's definitely an enthusiast perk. One that will undoubtedly prove incredibly profitable for them as that enthusiast crowd is known for shelling out the money for the things they want (i.e. PS Classics on PS4).
The major difference between the two is that the XB1 BC is focused on last gen software running at original fidelity presumably in an effort to stem the tide of 360 users migrating to the Sony ecosystem this holiday season whereas the PS Classics would be old school games possibly running at higher resolutions with additional features (trophies) aimed at the gaming enthusiast market that would remember and repurchase these older titles.
For the MS solution, the major problem is that they are trying to convert 360 owners but it's not a universal compatibility solution and, as such, would have very limited appeal to the mainstream 360 owners that haven't migrated to next gen yet. Thus, despite it being a very cool perk for those who have the system, it's unlikely to sway purchases in the way MS would like for it to. What's odd is they don't seem particularly eager or concerned with making their solution profitable for them. There doesn't appear to be any intention of bringing these BC titles for purchase on the XBL store (unless I'm mistaken) instead it's being sold as a value add for either previously purchased last gen games or new releases by way of bundling in previous gen entries in the series.
However, I fully expect Sony to approach their solution with profitability in mind from the get go instead of just being a value add for customers. I honestly believe they will have us repurchase titles we may already own due to the enhancements they are enabling on PS4. People would rightfully be upset but the people who love those games will still buy them. Nostalgia is a hell of a drug. That's been proven time and again. I don't really see this implementation spurring mass market adoption either. It, like the XB1 implementation, will sell some consoles to enthusiasts but won't make any meaningful sales outside of the core group that have already purchased these next gen consoles.
