Right, figured it out.
The 651's processing and the VT65's processing were interfering with each other and causing occasional artifacts with both BDs and DVDs.
I couldn't figure out why I couldn't engage 1080p Pure Direct on the VT65 when playing content through the 651. As it turns out, I needed to set the HDMI colour space to YCbCr 4:4:4. Once I'd done this I was able to engage 1080p Pure Direct on the VT65 and, like magic, the artifacts vanished and picture quality improved with both BDs and DVDs.
Technically, you should always set the colour space to 4:2:2, as less downsampling is needed; but switching to 4:4:4 had no detrimental effect on colour balance or timing. In fact, image detail and stability improved ever so slightly with BDs, while the difference was more noticeable with the 651 upscaling DVDs to 1080p. During DVD playback, with the 651 upscaling, it's still possible to select 1080p Pure Direct on the TV. I'm now very impressed with the 651's upscaling, which is better than my Panasonic 120. (The 120 was no slouch in terms of upscaling, either.)
The same is true whichever BDP you use (4:4:4 must be selected to allow for 1080p Pure Direct selection on the VT65.) This is a quirk of the TV, rather than anything to do with the BDP. I thought that might be useful for other VT65 owners to know.
Sorry if that got a bit dull and technical. The short version is that I'm now very happy with the 651. I can stick my 120 on Ebay, content that I bought the right replacement at a great price. The 651 is also quieter in operation, better built, and I have a five year warranty, so I'm covered if anything goes wrong. My only remaining gripe is the remote, which still feels a bit cheap and unintuitive for what's technically a £500 BDP.
Just goes to show, a minor setting tweak can make quite a difference. You have to live with tech and experiment to fully appreciate what it can and can't do.