I think we all know that these two instruments have quite a bit in common from a tonal perspective but have little in common from a playing perspective. Most, if not all, Gibson Les Pauls (especially the ones that haven't been weight relieved) weigh more which can get uncomfortable if stood up playing for a while. They are thicker bodied and have a shorter scale length, different necks too. The Custom 22's now only come with a trem bridge, different carve and better high fret access. You get a better designed headstock too with straight string pull and a better angle that is less susceptible to breakage. I think its a more modern instrument from a playing perspective. Neither is too far from the other sonically and both can perfectly functional for that 'double humbucker' sound that Les Pauls have built. Both can be interchangeable - for example you could use the LP seated at home but take the Custom 22 for a gig as its not as heavy and can offer a bit more with some in between sounds (Positions 2 and 4 of the 5 way) and the trem (if you have one).
They are also different enough that you don't have to get rid of one because you have no need of it. The LP with its independent tone and volume can lead to some more varied options when in the middle position - like having the bridge tone rolled down whilst keeping the neck tone rolled up, blending the two together using the volume pots. I understand why some may think the Cu22 is an alternative, trying to decide which is 'better' but whilst they can sound alike in some situations, both can also do things the other can't - more so if you have a trem bridge Cu22. I think that there are other PRS guitars that can be much closer to the LP.
Tonally, each can be 'better' as that comes down too personal preference as does playability and the different features, placement of knobs/switches etc can suit some people better too. The one thing that most can agree as better on a PRS is the string pull at the headstock but everything else is just preference. Between my 594 (vintage sounding) and Special 22 (more classic), both of which can offer a wider tonal palette thanks to independent coil splitting/tapping and the Narrowfield on my Special, I don't feel I need a Gibson Les Paul but I also would own the right one if the right circumstances allow.
In my opinion, comparisons like this are somewhat redundant as the guitars are quite different in enough areas that owning both isn't like owning 2 of the same instrument. Comparing a Gibson LP to a Eastman equivalent makes sense because both are incredibly similar and therefore does make sense to me to try and find which is the 'best' LP to buy where as deciding between a LP and Cu22 could come down more to the differences that each have, different body thickness and shape, different knobs/switches and layout of these, different Pick-up selection options, different scale length, different bridge etc etc rather than just the tonal differences.
Still I enjoy watching these....