Well, it's hard to compare older consoles similarly because they all used different architectures. The best comparison is probably the
Radeon HD 7770 vs. HD 7850, since those are the parts which most closely match what's actually in the consoles.
In practical terms, since the PS4's GPU is ~50% more powerful, that means if a game runs at 1080p30 on the PS4, it will only run at 900p30 on XBone. However, it's a little more complicated than that, because although the XBone's GPU is similar to the 7770, the 7770 should be backed by 2GB of comparatively fast GDDR5, but the XBone uses comparatively slow DDR3. The eSRAM is also comparatively fast, but it only provides 32 of the 2048 MB of "fast" RAM the GPU normally has access to.
That means that as a general rule, the XBone's performance will be
lower than the 67% on-paper performance listed. So a game that's 1080p30 on PS4 will probably be sub-900p30 on XBone.
Resolution drops to compensate for performance diferences seem most likely, since most gamers won't notice it as readily as missing particle effects, etc. but it's possible they might also drop certain effects the XBone particularly struggles with.