>Is it just the difference in daylight?

Yup, many plants have competing chemical processes triggered by light, darkness, and the wavelengths of light you tend to get in the morning/evening. As the days get shorter this balance shifts and trees leaves start to change color.

>If so how does the tree grow in the spring with similar day lengths?

Plants have a few different strategies here. Some plants do need cold here, as the cold will open access to a gene that was closed off during the fall, allowing them to flower when they days get longer again through a process called vernalization. It is pretty well studied because it occurs in the plant model organism (thale cress).

I am not much of a horticulturist, so I am sure that someone can probably give a much better overview of the different strategies.