I've hiked both and they are quite a bit different.
First and foremost, let's start with the obvious facts - the Wonderland is approximately 93 miles in length while the JMT is 215. The highest point along the JMT is of course Whitney, at 14,491 feet, which is 80 feet higer than Mt. Rainier. The highest the trail gets on the Wonderland is around 6,900 feet depending on the route you take. Some people do choose to climb Rainier at then end of a Wonderland trip, but this requires some mountaineering skills, although with a guide it can be a more of a long steady climb under ideal conditions.
You can pretty much camp wherever you want along the JMT, as long as you have a trail permit. On the Wonderland, a strict permitting system requires you to camp at designated spots each night. There are poles to hand your food at many of these camps, as well as a pit or composting toilet. I'd say except for a handful of cases, the campsites are not spectacular, but chosen to minimize impact upon the surrounding environment.
On purely aesthetic grounds, both have a lot to offer. The JMT is a more distinctive wilderness experience than the Wonderland, which is in a National Park within a short drive of the greater Puget Sound metropolitan population. The JMT is roadless and in the High Sierra, far from but a few population centers. Itfeatures granite, long climbs, distinct passes, deep gorges, plenty of water. The Wonderland follows the flanks of a volcanic, snow-capped peak, with plenty of healthy climbs and descents. On the Wonderland you travel through several riparian zones.
The weather along the JMT would be reflective of California during the summer - many lovely, warm days. There is the occassional threat of an afternoon thunderstorm, but this generally soon passes. The Wonderland is distinctive of the Pacific Northwest. It can be absolutely gorgeous - but just as likely you will experience rain and clouds unless you go at the height of summer or we experience a dry fall. It can be rather wet on the flanks of this mountain. Bring rain gear.
The Wonderland is often hiked in 7-10 days. That's not big miles per day - you certainly could do it faster, but it's a great trail to average 10 miles on or so per day and do a bit of exploration. It really depends upon your style. You can cache food ahead of time either by car or mail to ranger station at key locations along the trail. At most, you need to carry 3-4 days worth of food. You can even schedule a day at one of two lodges (Longmire and Paradise, which is technically a bit off trail) to break up the hike. There is also a small store in Longmire and a restaurant at Sunrise where one can by a snack or meal.
The JMT typically takes around three weeks to hike, although, you can certainly hike the entire thing much faster (particularly for thru-hikers, who are by that time in very good shape). The limiting factor on mileage is adjusting to altitude, the number of passes you can do in a day and resupply strategy. You are required to carry a bear canister in sections of the trail. Resupply will force you off the trail, in some cases, a fairly good number of miles (for example, if you choose to go into Independence or Bishop). But some welcome the respite by going into town and resupplying, getting a motel and a break. In any case, you will be carrying more days worth of food on the trail
But what are the hikes like? The expansive views along the JMT, particularly on the high passes, are beautiful. You can see peaks for many miles, culminating in the climb up onto Whitney, the highest Peak in the continental United States. You hike among great granite cathedrals, an inspiring sight to behold. Frankly, I'd argue that Bighorn Plateau is among my favorite spots on any trail anywhere, for it's panoramic views. You descend into forests, green lush meadows only to climb again amidst impressive waterfalls, large snow fields (depending on when you go, there can be very little or miles of snow to cross), and rushing creeks. It was dubbed "The Range of Light" for a reason. I had high expectations, and it honestly exceeded them.
The Wonderland is distinctive because of its presence over Western Washington. Unlike Mt. Whitney, which stands in concert with many high peaks (and for some time, it wasn't clear that Whitney was the tallest), Mt. Rainier dominates the skyline. On its flanks you will pass through many riparian zones, gaining an appreciation of the breadth and scope of one mountain. The overwhelming sense you will be left with at the conclusion of the trip is the distinct differences in climate and terrain depending where you stand on its flanks. You have the pie crust of the west side, a near-constant up-and-down through volcanic soils and deep, glacier-fed rivers. While the sheer scope of the High Sierra is the beauty of the JMT, the intimacy of hiking one lone mountain carries with its own brand of pleasure. You hike among the hemlock and Douglas fir of the west, experience glaciated valleys, and climb to gorgeous mountain meadows. Water is available in abundance on both trails.
In terms of toughness? I'd say both are physically demanding. The JMT gets the nod because of elevation. However. the weather, particularly in mid to late summer, is characterized by dryer weather and lower humidity than its Washington brethren. On the Wonderland, you will have higher humidity due to the proximity to the coast, and probably experience more changes in weather.
If you want a wilderness experience full of high passes, choose the JMT. If you want a mountain experience, with less time/mile pressure and much more simple logistics, choose Rainier.
Finally, I'd propose at alternative...if you were to hike the JMT and wanted another longish hike, I'd propose hiking from Stevens Pass in Washington to the Canadian border along the PCT. This is among my favorite stretches of trail anywhere, cutting through the Glacier Peak Wilderness, North Cascades National Park and the Pasayten Wilderness. It's about a 190 miles long.