Terry:
I've played them both (and played them within the same two week period last fall). Here are my thoughts:
What are the similarities and the differences?
General similarities - Both are links-style courses with generally wide fairways and big greens. Both have areas of longer grass between holes, with CB having sandy waste areas in parts between the fairways and the gunch. Both are walkable (albeit harder walks) and have caddy programs. EH and CB have plenty of quirk and have several holes with fairly severe green contours (although it sounds like some of these contours at CB may be softened).
Differences - CB's site is comparable to the side of a soup bowl (carved out of an old quarry), while EH is on rolling land (think undulating farm land). The EH site feels much bigger than the site for CB, and its routing has a meandering feel that you don't find at CB. As mentioned above, CB utilizes the sandy waste areas as a hazard, while EH has smaller well-placed fairway bunkers in places. EH seems to have been laid on the land, while some of the contours at CB were obviously created. With a few exceptions, the holes at CB are generally smoother (both in elevation and fairway contours) from tee to green than the holes at EH.
Is either properly described as a links, in any way shape or form?
Both have firm turf and can have wind (although not as much wind as you'll find on a true seaside links). Perhaps it was the conditions I had when I played the courses, but I thought that tee balls ran a bit more at EH than at CB. Both courses give you the opportunity to play running shots in places, but also require aerial shots on other holes. If I had to pick one of the two as the course where you'll get more movement on the ground, I'd go with EH.
Which is a better golf course and why?
This is going to come down to personal preference. I like the feel and play of EH better than CB. Both are a strong test, but I think EH may be a bit tougher for the pros due to its ability to challenge the player when they are out of position. See my hole group comparison below.
Which is better suited for major championship golf and why?
I think they'll both be a good test. However, they may both be better match play courses than stroke play. It'll be an interesting exercise to compare the relative successes of the amateur championships to what happens when the pros get there. If I had to guess, I'd say that CB is more susceptible to being bowled over by a big hitter than EH. From a spectator standpoint, the vistas created by the land at CB will create great sightlines and allow patrons to take in a fair bit of action from one vantage point.
Which is more worthy of a cross-country trip to play?
You can't go wrong with either. If you're playing additional golf on this hypothetical trip, EH might win due to the proximity of Lawsonia, Kohler and the quality courses in Milwaukee. If you're combining CB with a Bandon/Oregon trip, you'll have a bit more driving time, but the quality of golf will be pretty high.
Here's a brief comparison of the two courses by similar hole types (some of these holes may be reclassified depending on how the USGA decides they should play - i.e. the 4th at EH which was played at around 300 yds for a few rounds during the AM):
Par 3's -
CB (3, 9, 15, 17) - I didn't think the par 3's here were the strongest group. There's a bit of interest around the greens (especially on 9 and 17), but there was not a ton of variety in the types of shots required. With the exception of 17, I don't see any of these being a tough par for the better players.
EH (6, 9, 13, 16) - A slightly more diverse group than you find at CB (with the long 6th being a good test). The 9th at EH and the 15 at CB would be the most similar of the 8
I'd give the edge to EH.
Short Par 4's -
CB (2, 10, 12, 16) - 10 and 12 are the standouts here. Will be fun to see how 12 plays, as I find it hard to imagine many not having a go at it.
EH (2, 11, 12, 15) - If they play 4 at 300, EH wins in a landslide. Otherwise, its a pretty even race with EH winning by a neck. The table top green on 2 will present an exercise in control.
EH wins here.
Long Par 4's -
CB (1, 5, 6, 7, 11, 14) - The backbone of the course, with 5, 7 and 14 being the best. I did not play the tips at CB, but from the up tees all of these holes offered plenty of challenge off of the tee (ie - avoiding going left on 1, setting up a good angle on 6, taking on the corners of 7 and 14). With 7 being softened to be more receptive on the approach, a bit of the bite around the greens is going to disappear from this group.
EH (3, 4, 5, 8, 10, 17) - The new renditions of 3, 4, 5 and 10 may not be better holes for the paying customer, but they certainly are harder. The punishment for a miss at EH (especially on the long par 4's) is a bit more severe than at CB. Where a miss on 7 or 14 at CB may require a recovery from the waste area, the same miss at EH will require a shot from the long grass (if not a lost ball).
I give the edge to CB here due to the variety of the holes and the diagonal nature of a few of the tee balls.
Par 5's -
CB (4, 8, 13, 18) - No real standout hole, and perhaps the hole (8th) that gets the most criticism at CB. 4 and 13 feel fairly similar, and there is nothing about any of these holes that pops out as exceptional.
EH (1, 7, 14, 18) - A love 'em or hate 'em group. 1 will probably be a cake walk for the pros, although 14 and 18 may give them fits.
EH wins in this comparison, but as I said, its really a matter of personal preference.