If you want to stay at a "brand name" 5-star hotel chain in Kyoto there's really only two options: the Four Seasons (FS) and the Ritz Carlton (RC).  The RC was the first game in town and the Four Seasons is slightly newer than RC. (There are other Kyoto options of course, but Suiran is in a very different area of Kyoto and a high end ryokan is going to be a different experience.  And Aman Kyoto is forthcoming.)  

Design-wise, I see a palpable philosophical difference between the FS Kyoto and RC Kyoto.  Although they both have plenty of Japanese design elements, the FS struck me as more "traditional western luxury" because it uses a lot of light marble, shiny floors and has a big, open bright lobby.  The RC Kyoto feels more Japanese in terms of using a lot of wood and patterns and the lobby feels more intimate, dark, dramatically-lit and lounge-like.  The same design aesthetic applies to the rooms.  For example, the bathrooms highlight the biggest differences: the FS has marble everything while the RC is mostly wood and black stone (especially the RC shower with wood slats on the floor).  In the common areas the FS design seems quite a bit more bland while the RC was more dramatic (esp. lighting effects).

Room

I stayed in a "Premier Room -- Kyoto View" room at the FS.  There are only three rooms in that category (room number 417, 419 and 421), and I booked two of them for 5 nights :-)  The view of the temple was nice but a bit boring and basically dark at night.  In retrospect, I suspect a garden view might have been more interesting (but from reviews it seems like a fair number of the garden view rooms have a partial or poor view where most of the view is a road or wall).

Ignoring the aesthetic / decor differences, on balance, a room at the FS Kyoto is probably better equipped than a comparable room at the RC Kyoto: larger, decked out with more features, etc.  The FS (non-suite) rooms are also a bit larger than RC, especially the entry level category  All standard FS rooms are ~570 sq ft while the entry level RC are ~483 and the Luxury rooms are ~560.  [I stayed in the Luxury category room at the RC.]  

Although it doesn't look like it on first glance, the FS room has a lot of closet storage space.  It's been a while since I stayed at the RC, but I feel like it was more closet space than our RC Luxury room.  There's also more of a real desk for a laptop at the FS.  

Some musings on room features:

• Our room didn't have a balcony but it did have a door that could open plus a little retractable bug screen.  
• The shower and tub area has a nice trick where it can turn into a steam room by folding the shower glass doors outward to block the entrances to the bath chamber.  
• I also like the fact that the bathroom can completely close off from light with opaque sliding doors.  They also lock.
• As a small downside,  the tub / shower area is sunken and creates a toe-stubbing hazard.  
• In room bath amenities were Lorenzo Villoresi Firenze.  They were unmemorable.
• The separate toilet room has a coat hook.  I like that!  I also like the fact that you can raise the seat with the Toto control and not have to lean over... just press a button.
• The chairs in the room were slightly awkward for in room dining.  One person often ended up having to use the ottoman as a stool.

The FS has an in-room tablet for ordering all kinds of things.  E.g. you can have them send different pillow or ice to the room via the tablet.  They also have no in room dining menu except for the a tablet.  In general, I really dislike it when hotels do this because it makes it hard to browse the menu at a glance.  You have to click on a category, scroll through that set of items, click on another category, etc.  It also hides things.  For example the chocolate cake dessert is under "express menu" category... not desserts.  The delicious Bolognese pasta entree is under the "express menu"... not entrees.  The tablet menu UI was far far better than the terrible tablet at the Peninsula Paris, but it would still crash every time you navigated to the Spa section.  Sigh.

Service

The 4S Kyoto service was mostly positive for me but I felt let down on housekeeping being out of my way (more on that later).  

My pre-arrival concierge experience was great.  I booked a few events via email and everything was very smooth and responsive.  A few days before check-in I got an automated email to indicate my arrival time and noticed that they suggested contacting the hotel via Facebook messenger.  They were super responsive over FB messenger and I used that consistently; while riding the bullet train from Tokyo, I was even able to arrange a van with a meet and greet at the rail platform using FB messenger on my phone!  Every hotel should do this as far as I'm concerned.  It also helped with the language barrier because text avoids pronunciation issues.

At the hotel my main service gripe was housekeeping timing: they were always in the room or coming by when we most wanted to relax in our room after sightseeing. I like it when housekeeping is like a magic invisible team that sneaks in whenever you weren't around and you always came back to a fresh room.  At the 4S, we we always puzzling, "hrm... they still haven't cleaned things at 3:30pm when we've been gone since 9am?" And turndown rang our doorbell at 9:45pm when we were already in bed.  On day 4 of 5 we just started pressing the "service room now" button when we left and that mostly worked but I felt like I shouldn't have had to do that, especially when a lot of the events we were going to were arranged by the hotel concierge.

Also, pretty much whenever I called the front desk or room service it would ring for minutes and then the operator would tell me they were busy and would have to call me back later.  

In the room they were constantly bringing us fruit or Japanese sweets or little tiny gifts.  It was a nice touch.

Location

I don't really care much for the location.  I've stayed at the Hyatt Regency Kyoto before the RC was open and it's literally within a block of the 4S.  Some people don't like being close to Gion or touristy areas but I preferred the RC location because I like to have options to grab something quick and cheap like Starbucks in the morning.  Around the 4S nothing is open until like 10am (save a 24 hour McDonalds) and your on property options are limited and overpriced.  Many things I'd like to walk to are 20-30 mins from the FS vs. 10-15 from the RC and my traveling companions are not as patient so that mostly meant taking a cab vs. walking.  :-)

Also, note that from 8am to 9am every weekday you can't take a cab from the front entrance.  You have to go to the residence side of the hotel. They close the street while school children are coming to the school right nearby.
 

Dining

I only tried the Brasserie and in-room dining (which is mostly from the Brasserie as well).  I'd say the food was serviceable on average but overpriced.  I never tried breakfast and I mostly ordered western food. I specifically enjoyed the Bolognese entree and hated the Caesar salad.  The caprese with mango and the foie gras appetizers were good.  The Japanese beef curry rice dish had fantastic fatty medium rare slices of steak but the curry sauce was pretty one-note, flavor-wise.

One notable bar / lounge option is the tea house in the garden.  They serve afternoon tea 2-5pm and then champagne and sake from 5-9pm.  You can order champagne in the afternoon too though.  I didn't try tea but was in there for champagne a few times.  It's a nice peaceful environment.

Spa / Pool / Gym

The gym is quite well equipped for a small luxury hotel.  It was much larger than the RC gym.  The FS also has men's and women's saunas and a hot and cold tub in each respective changing room.  The changing rooms also had relaxation area alcoves with daybeds in them.

I didn't use the pool but on my one walk-through it looked to have some interesting pool chairs that were partially submerged in the pool, plus some cabana like seats around the pool. 

Overall

Overall, the Four Seasons has very good in-room hard product and large entry-level rooms.  Compared to the RC, the FS's public spaces were a bit bland design-wise.  And the service was mostly good excepting the annoying housekeeping timing.  The concierge was especially responsive (and especially via FB messenger).  I don't love the location.  A good stay, everything considered.