What are the major differences in how different sizes of company perform?

Huge differences. Small companies are typically more agile, they move faster, they have less layers of management. But you probably won't have a highly defined job role. A small company means that somebody has to do the work regardless of whether it's in their wheelhouse, so you are more likely in a small company to be pulled into an area of the business that you're not terribly comfortable with. This is a double-edged sword - I was forced into network engineering due to small company staffing, and I despised the idea of being thrown into a river and being forced to swim. But the knowledge I gained has served me well in my career. A larger company will typically have more defined roles - this is Department A and they take care of 1, 2, and 3. Department B takes care of 4, 5, and 6. This can be both good and bad - you know exactly what you're responsible for, but it can get a little boring when you're stuck doing the same thing every day.

I agree with Patrick and would add this: move to a smaller company for a bigger title; move to a larger company for bigger money. A small company tends to lack structure which means titles and tasks are very flexible--it means you can find yourself doing things you'd never be asked to do in a larger company. A larger company expects you to have very specific skills aligned with your job title and you won't deviate much from that job description.

In general, if you're not sure of your interest and skills, a smaller company gives you much better growth opportunities. If you have strong skills in a specific area, a larger company offers more money and less risk.