The differences are not always cut and dried, but here is the general idea. Each bike involves some level of compromise. There is no free lunch in cycling any more than there is in anything else, so bicycles tend to be designed for a specific purpose.
A road bike is designed to ride its best on pavement. It will mostly have very thin tires that can take high pressure for the least rolling resistance. Though it is not a race bike, it will be fast on pavement. Some compromises were made from a race bike that make it some what more comfortable than something built for all out speed. They also perform pretty badly on just about any surface but pavement. Those narrow tires sink in and go no where unless its a hard surface under the wheels. Usually you will see "drops" for handle bars. It allows you to shift hand positions from time to time.
A hybrid is just that, a bit of on road and a bit of off. You usually sit a bit more upright than you would on a road bike. It makes it a bit more comfortable than a road bike, but not quite as fast. Gearing is usually close to a road bike, but a hybrid can do something on which a road bike would struggle. A hybrid would be much more at home off the pavement on some very light trails. Hard packed dirt or some other solid surface a hybrid will usually handle just fine. It is not the off road bike a mountain bike is or can be, but it can handle some well maintained trails that wouldn't work on a road bike.
The touring bike is a bike that is designed for long trips. It usually has a stronger frame that can carry extra weight for things like camping gear, extra clothes ect that you would need on an extended trip. It also tends to weigh more. This bike is designed to be ridden for long periods of time. Speed may be compromised a bit for comfort if you are going to be on the bike 8 hours a day. Many times they are made of steel, so quick welds can be made on the road and cracks ect that might occur along the way can be fixed at the local welding shop. A carbon fiber bike a crack is a big deal and could take weeks to get the bike repaired. That on a tour is a big deal.
You would also expect to see the gearing a bit different. Many touring bikes have gearing more to get up that big hill rather than to go at a blistering speed on the flats. Speed is not the priority on a touring bike. Comfort and durability are more important. I read a blog a while back though that a guy that did a lot of touring, kept an extra cassette with different gearing in his bag. When he came to the mountain or hilly areas, he had the climbing cassette on the bike, and the other in the bag. Once he got through the hills, he would pull his flat lander cassette out of the bag and swap them on the road. Gave him more speed on the flats to make better time. The trade off is he had to carry the extra weight.
The answer is that it depends on what I want to do and where I want to ride. Few realize that a bike is a tool that is designed to fit a particular type of riding. You can drive a nail in with the back of a hatchet, but a hammer works much better and you are not as likely to get cut. I do not even own a hybrid. I have a mountain bike, but I don't get to ride the trails that often so it is on the trainer most of the time. What I ride most is none of the above. I have a road bike, which is the fastest bike I own. It's nothing fancy, a Fuji Newest 3.0, but its a quality bike that serves me well. I ride it once and a while and keep it mainly because a friend that's very special to me sold it to me a few years ago. It was the classic bike purchase mistake from a size perspective. It was just close enough to fitting that on a short test ride it felt fantastic, but after a few miles I began to hurt. Tried every adjustment I could find trying to make a bike that didn't quite fit, fit. I fooled with it long enough I hurt a knee. Then that had to heal. After a couple of years, a different seat post and an unconventional way of twisting the handlebars I finally got it to fit. I ride it once in a while, but not too often. Yet I can't get on or work on that bike without thinking about Mariah and that makes it a good day. That friendship alone could not have been bought and even if it could have, at the price of the bike it would have been a bargain even if I never rode it. I have no idea what Performance Bike pays her, but it probably isn't enough.
The one I ride the most is my Sun Ray. For a new rider here is a search engine exercise for you. Its a compact long wheel base, semi recumbent bicycle. It was meant as a cruise type of bike, and really never intended to be that fast. It's steel and somewhat heavy, and when I added rack and panniers it got even heavier. Why this one? It is the most comfortable. You do not have a saddle but a seat that resembles a stadium seat, that is upright. (a full recumbent you would have a full backed seat that is laying back for less drag and more speed) and a bit like pedaling a lawn chair. I changed a few things on it. I changed the tires so they can take 100psi to reduce rolling resistance on the pavement. I put a different cassette on the back that was close to the gear ratios on the Fuji. I gained a little more speed, but lost the really big grandma in a walker gear for climb. It was a trade off I was willing to make because if I really needed that gear to get up a hill, it was slower than walking it anyway. That slow who ever would be laughing at you for walking up a hill would be laughing at you for going that slow, and that assuming I gave a rats behind about what they thought. (which I don't)
The Sun Ray has its down sides. Nimble it isn't. It's length makes it feel like you are riding an 18 wheeler of a bicycle. If you need to make a sudden maneuver in traffic, the Sun Ray will not do it nearly as well as a standard upright road bike. There is a possibility that I could encounter a need to make an emergency maneuver that on an upright bike could be done, but not with the sunray and what ever it is I am trying to miss I end up hitting. I ride with that in mind. The Sun Ray is a stable bike, that is until it starts to slide. When things start to get out of hand on the Sunray it happens in a hurry and it is much harder to catch than on an upright bike. Wet roads require even more care than they do on an upright bike.
There is a point to all of this. There is no best bike. A bike that could win a time trial on the Tour de France would be ill suited for a world wide tour. If fact they don't even use the same type of bicycle for every stage of the Tour de France if I recall correctly. Pick the one that suits your needs, fits you and right for where you want to ride and you will have a good time.
Like Photosbymark says, it really depends on what you are trying to get out of the ride. But at the same time, you can do almost anything you want with any bike. so while you'll find that a lot of us on this forum have multiple bikes set up for different tasks, you can do it all with one, if you really want to. I will say that if you are going with a single ride, a hybrid gives you the flexibility to ride on road and off, and even some longer distances if you are ok with a long ride in a more upright position. Most hybrids will take racks for hauling a load too. If you want similar flexibility with a more forward position, a cyclocross (cx) bike gives you a lot of options too.
Have 3 bikes: 2 road and 1 hybrid. For what I do down here, the road bikes get the most use, even though some trails have just opened up around here, perfect for my hybrid (running 700 x 38 tires, smooth tread as I don't need a full on mountain bike nor the knobby tires).
I can only afford one bike and it is a hybrid. Live on a road that leads to an industrial park. If I encounter two trucks going in opposite directions at the same time I hit the shoulder. I don't argue with trucks.
Yeah they are awesome for riding in the rain! Fast too. A couple months ago a bunch of them raced in 1 hour time trials on a wooden velodrome and were getting between 36 and 39.7 mph average for the hour.
You can check out bluevelo.com where I bought mine from or in the netherlands at http://www.velomobiel.nl/
They've been building velo's overseas for over 10 years now and just 3 years in canada. I'm shooting for a 3:45 century this year in a time trial with full race kap.
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