Three Things that Make Polymers Different

There are three ways in which polymers will act differently from small molecules, which are usually named chain entanglement, summation of intermolecular forces, and time scale of motion.

Chain Entanglement
Most polymers are linear polymers, namely, they are molecules whose atoms are joined in a long line to form a huge chain. Now most of the time, but not always, this chain is not stiff and straight, but is flexible. It twists and bends around to form a tangled mess. The chains tend to twist and wrap around each other, so the polymer molecules collectively will form one huge tangled mess.
When a polymer is molten, the chains will act like spaghetti tangled up on a plate. If you try to pull out any one strand of spaghetti, it slides right out with no problem. But when polymers are cold and in the solid state, they act more like a ball of string. Trying to pull one strand out of this mess is a little harder. You're more likely to end up making a big knot! Solid polymers are like this. The chains are all tangled up in each other and it is difficult to untangle them. This is what make so many polymers so strong in materials like plastics, paint, elastomers, and composites.

Summation of Intermolecular Forces
All molecules, both small ones and polymers, interact with each other, attracting each other through electrostatics. Some molecules are drawn to each other more than others. Polar molecules stick together better than non-polar molecules. For example, water and methane have similar molecular weights. Methane's weight is sixteen and water's is eighteen. Methane is a gas at room temperature, and water is a liquid. This is because water is very polar, polar enough to stick together as a liquid, while methane is very non-polar, so it doesn't stick together very well at all.
Intermolecular forces affect polymers just like small molecules. But with polymers, these forces are greatly compounded. The bigger the molecule, the more molecule there is to exert an intermolecular force. Even when only weak Van der Waals forces are at play, they can be very strong in binding different polymer chains together. This is another reason why polymers can be very strong as materials. Polyethylene, for example, is very non-polar. It only has Van der Waals forces to play with, but it is so strong that it's used to make bullet proof vests.

Time Scale of Motion
This is a fancy way of saying polymers move more slowly than small molecules do. A bunch of small molecules can move around a lot faster and a lot more chaotically when they're not all tied to each other. Tie the molecules together in a big long chain and they slow down. This slow speed of motion makes polymers do some very unusual things. For one, if you dissolve a polymer in a solvent, the solution will be a lot more viscous than the pure solvent. Actually, measuring this change in viscosity is used to estimate polymer molecular weight.