What is the difference between the albuterol in my new nebulizer and the albuterol in my rescue inhaler?

Nothing really. The primary difference is the device used to deliver the albuterol. The medications are the same and if the device is used correctly, both techniques should yield the same beneficial results.
Very little. Studies have shown that albuterol administered by nebulizer has the same effect as that given by a rescue inhaler/metered dose inhaler. Patient preference really guides the increased use of the nebulizer treatments in recent years but the effect is the same.