A touring bike might be thought of as a more-practical roadie. You can attach fenders and racks, as Hack pointed out. The racks give you the ability to carry a whole lot in panniers, or less attractively by strapping stuff right to the racks.
As a result, you'll find touring bikes are geared lower to allow you to schlep your cargo up steep grades. This obviously costs some the top end gearing, but it hasn't been a big deal for me - I can still exceed 40mph downhill with a couple of cogs left over in back. I ride a 2011 Novara Safari.
Because they're made to carry heavy loads, many touring bikes also come with heavier-duty brakes. Mine came stock with highly effective rim brakes, but has all the mounts needed for discs. It's cool to have the option of using either. Not all touring bikes are like this, though.
Again due to heavy loads, most touring bikes come with heavier-duty, more robust rims - double-walled and with heavier-gauge spokes. The rims (and bike frames) are often also designed to mount wider tires.
That brings us to tires. Touring-specific tires are, again, designed to handle heavy loads and to go long distances without flatting. As a result, they tend to be heavier and sport more rolling resistance than racing tires. Between the added tire width and toughness, a set might be just the thing to fix the tire cutting issue referenced in he OP. That's gotta drive you nuts
I still recommend rolling with heavy-duty tubes loaded with slime or its functional equivalent. When it comes to slime and presta valves, a few manufacturers DO make presta valves with removable-valve cores so slime can be added.
You do not, of course, have to ride a touring bike loaded. It's just nice to have the option to do so. It's wonderful to walk into a store in cycling gear and buy something you couldn't possibly fit on a regular bike - you see which clerks are paying attention vs. working on auto-pilot. The facial expressions from those who do notice are often quite amusing
In short, touring bikes rule, though I may be a bit biased since I'm not a racer-type
If you are the racer-type, I might point out that building the muscle to ride a heavier bike does great things for your ability to accelerate and handle a lighter roadie when the time is right.