There are many styles of multitool too, e.g. cards and keyrings but the two most popular ones are the butterfly-style multitool, of which Leatherman is perhaps the most famous manufacturer, and the swiss army knife, for which Victorinox is best known.
Butterfly-style multitools tend to be better for heavy-duty use, e.g. DIY, automotive, gun maintenance, whereas swiss army knives are generally better for travelling, fishing, climbing, backpacking and general use.
Perhaps the main reason for choosing the butterfly-style of multitool is that they have usually have pliers, often with some form of wire cutters built in. Some swiss army knives have pliers too but they tend to be less solidly built and give less gripping power. The handles on the butterfly-style are usually solid metal and hence heavy. Swiss army knives tend to have plastic handles and so are notably lighter given similar tools.
The shape and functioning of the main blade also tends to be different on the two styles. Swiss army knives usually have a non-locking blade that is good for light cutting tasks, e.g. cutting string. However, the blade can fold back onto your fingers so care must be taken. Butterfly-style multitools tend to have locking blades which makes them safer for heavier use as the blade cannot fold back. Bear in mind that such a blade will be considered a fixed blade which are illegal to carry in some countries (e.g. the UK unless you can prove you have good reason). Some butterfly-style multitools need to be opened before the blade can be accessed, making it less accessible than that on a swiss army knife.
Swiss army knives often have personal grooming / first aid tools like tweezers, nail files and toothpicks. Nowadays some have torches, usb drives, and even mp3 players.
Some butterfly-style multitools have interchangeable and replaceable tools. Swiss army knives generally do not.