The difference between rats and mice isn't just "rats are bigger," although that is true. Rats are usually five to 10 times bigger than mice. They are, however, both rodents and have several similar characteristics, such as characteristic teeth, whiskers, poor eyesight, and heightened other senses.
The differences are striking, however. Mice, for example, are curious in the extreme and will explore everywhere. Traps are much more effective against mice, rather than against rats, who are much more cautious than their smaller brethren. If you put out a rat trap at all, a rat will avoid it at all costs until it becomes familiar with it. Therefore, a good strategy for trapping rats is to place unset traps without bait in the paths they usually traverse. The same holds true for kill traps and Havahart® traps. Once a rat "knows it's there," the rat will begin to treat it as a part of the landscape and will cease ignoring it. That's when to bait the trap.
The dangers posed by mice and rats are also different. Mice seldom carry disease, but they poop everywhere, and their poop can infect food and other consumables in the home. Rats, on the other hand, can carry several terrible diseases, such as leptospirosis. This disease can cause liver and kidney damage, and you can get it just by cleaning up rat pee and poop. Their bites can spread Lymphocytic choriomeningitis, which is a form of encephalitis. You haven't seen it much in the last 750 years, but the bubonic plague is a rat-borne disease, too, even if it's the fleas on the rat that carry it and not the rat itself.
Remember, even though mice don't usually carry disease, it's never a good idea to approach or handle any wild mouse or rat. Rodents that are cornered will fight fiercely and bite accordingly. They also reproduce very quickly, so a "couple of rats" can turn into an infestation almost before you know it. Because of the danger of disease, it's always advisable to engage a professional to get rid of them.