How does the logistic model of population growth differ from the exponential model?

Firstly models are just predictors, they are not exact models.

Exponential growth models are good when populations are small relative to the amount of resources available. For example a small number of rabbits are released into a field or a small number of fish have been released into a lake. There will be plenty of resources for them to multiply as quickly as they want.

Eventually though, when populations get larger, it will have to compete with each other over the resources and they will not be able to multiply as they have previously. This is where the exponential model breaks down.

The logistic function tries to compensate for this with the carrying capacity. It takes into account the limited resources and slows down the growth as it nears the carrying capacity.

An example of this is a mature forest. Why doesn't the forest just keep growing and growing? Because there is a limited amount of water, sunlight, and nutrients in the soil. There are other factors, but this is the main idea. So only so many trees can grow before the forest stops growing.

Easy reading, plain English on this website: http://www.dummies.com/how-to/content/the-environmental-science-of-population-growth-mod.html