A gazebo is an outdoor garden building of open-sided, usually wooden construction. It's essentially just a stylish way of saying porch that you can put on your house if you want to get away from the elements while still being somewhat protected.
You wouldn't find anyone partying inside one, though. They're typically too open for that. A gazebo might be used more for enjoying some time alone than it would be for social gatherings or events. Gazebos are also backyard structures - they don't look like they belong in formal gardens.
What is a garden pavilion?
A pavilion, on the other hand, is meant specifically to compliments formal gardens and often sits at one of the garden's entrances. It can be a single floor or two stories and is an elegant way to add some shade and extra seating while looking like you've been there for years.
The pavilion at Versailles, for example, was built in 1663 and still stands today - it's even open to the public (for a fee). The structure took four years to build and required workers from all over Europe due to its massive size (it spans over 65 acres!) That should give you an idea of how ornate pavilions really are.