Full-time employment means that you are paid a monthly salary for a given amount of teaching hours per month. This salary should be paid despite public holidays, school activities, and other random class cancellations.
If you are teaching at a high school, full-time may also mean having to be at school for assembly, and having to join in extra-curricular activities. Full-time jobs may also bar you from doing outside work during school hours even if you don't have any classes. Despite the full-time label, some schools will not offer 12-month contracts, i.e. you will not get paid during the summer months. Universities generally allow full-time teachers to come and go as they please, and usually give 12-month contracts.
Part-time work could also be described as freelancing. Part-time work is paid per class taught at an hourly rate. The upside is that the teacher only has commitments to the classes he/she teaches and not to any other aspect of school life, with the teacher working as little or as much as he/she wants. The downside of course is that a part-time wage is not very stable considering the amount of public holidays, and class cancellations that are guaranteed to occur throughout the year. For example during December it is possible to lose up to 40% of your income due to Democracy Day, the Kings Birthday, Test Week, and New Year. Overall, part-time work is good for newbies and retirees in need of something to occupy their time.