Some thoughts:
LR, is catalog only, (which I do not prefer), C1 allows either and I always be a session user.
C1, to me has a more sophisticated color selection tool, especially in version 8.x. C1 allows a local adjustment layer with color corrections, and you can pick the color very easily. Where as LR has color adjustment in the adjustment brush I do not like the method as I find it harder to work with. You also can now have local layer white balance control in C1 vr 8.x. Outstanding.
C1 to me is the best by far for any Phase One files, even the older Phase One backs, P25, P45+ etc. In fact the latest version of C1 has allowed me to pull some excellent results from my older P45+ images.
C1 loads with less color profile options for non Phase One cameras, usually just one profile, LR will offer several and it's important to try many of them IMO.
LR has a history, C1 does not. The LR history stays with the catalog, so as long as an image is in the catalog, you can see the steps you used on it. If C1 would offer this, I would be in processing heaven, but they seem either to not understand the need or can't do it. I do a huge amount of processing in LR or C1, very little in CC now and the history of work is just a huge advantage, at least to me.
C1 allows you to turn on and off a local adjustment layer (which is a huge advantage), LR, all adjustments either on or off, you can't just turn one off to see the effect (at least I can't do it)
LR with latest version has very powerful tool in the "auto mask" and it can really save processing time. I can't figure out why such a great selection tool can't be incorporated into CC? C1's auto mask IMO has a lot to be desired at this stage.
LR has a excellent print module, I use it exclusively as I don't want the extra cost of a RIP, C1, I have never printed from.
LR allows plugins, from tools like Topaz or Nik, I don't believe C1 allows this.
LR has an excellent HDR merge function with latest version and the Photomerge feature is very good. KEY is that you can merge the files while still in LR and they are seen as a large raw file. Wonderful tool. And the merge function is excellent from what I have seen.
C1 has the best LCC tool for a Phase One back, hands down, if you are using a tech camera, you are pretty forced there. LR has a method, but I tried it and did not like it.
LR seems to still get bloated (i.e slow) when you are working a file with a lot of adjustment brush settings, might just be my computer? 32GB of ram, 1GB of video GX470 nvidia? win64 i7
C1 DOES not like to work from a raid device, at least a Drobo. Loads images way slower and at times will lock up. LR does not seem to have this issue.
Many seem to not to use output sharpening from either tool, I do, works for me fine.
I would give the fringing ability edge to LR, as it has a more robust tool set, where as C1 is pretty much a check box.
C1 loads Phase images with a very strong "film curve" that has a lot of contrast and saturation, which I tend to like, but there are times I will back off to the linear curve for starting, especially on images in low light. LR on the other hand seems to load the images more neutral.
I feel both LR and C1 have excellent shadow and highlight recovery tool sets and these are allowed in the adjustment brush in LR and the local adjustment layer of C1.
These are the most obvious I can think of from using both tools side by side over the years.
Paul