As for a gazebo, people must be familiar with it, for it can be seen everywhere, such as park, garden, path to travel, etc. While people may not know pergola well, it is often seen at a garden after all. Both of the two can be regarded as the shelter for people to have a good rest or enjoy the cool in summer, which is seems to be the similarity of the two. In fact, pergola is a type of gazebo. However, more people may not know the difference between patio gazebo and pergola.
We can easily find the basic definition from wikepedia: gazebos are freestanding or attached to a garden wall, roofed, and open on all sides; they provide shade, shelter, ornamental features in a landscape, and a place to rest. Some gazebos in public parks are large enough to serve as bandstands or rain shelters. A pergola, arbor or arbour is a garden feature forming a shaded walkway, passageway or sitting area of vertical posts or pillars that usually support cross-beams and a sturdy open lattice, often upon which woody vines are trained. As a type of gazebo, it may also be an extension of a building, or serve as protection for an open terrace or a link between pavilions.
Gazebos can be either freestanding, or attached to a garden wall, and they are, traditionally, open on all sides, such as the Victorian gazebo, in England, shown above. In terms of their purpose, then, they provide shade and shelter, and larger ones, e.g. in public parks, can even be used as bandstands. However, some are purely ornamental.
Some of pergolas may be attached to the side of a house, while others are freestanding. They can be quite small, or they can be long, covered walkways, sometimes connecting two pavilions. Although traditionally made of wood, you will also find metal ones (e.g. wrought iron) and ones where the supporting pillars are made of brick.
From what have mentioned above, we know gazebo is very similar to pergola. Differences between gazebo and pergola reflect in shape, size, basic purpose, position of building, etc. Of course, if you are willing to plant vinifera on the pergola, I believe vinifera will be glad to grow up on the branches. When resting under the pergola, you can eat the delicious grape at the same time, saying a stone kills two birds, why not?