Every time when I create a script to manipulate vector (point, line and polygon) in python I meet this doubt.
What is the difference between bounding box (bbox), envelope, extent, bounds?
I think you'll find there is a bit of overlap with these definitions. They're all very similar, in my opinion. However, ESRI has a glossary of GIS terms, so I just looked them up. The definitions are similar or identical to the wiki GIS glossary as well.
[map display] The rectangle, aligned with the coordinate axes and placed on a map display, that encompasses a geographic feature or group of features or an area of interest. It is defined by minimum and maximum coordinates in the x and y directions and is used to represent, in a general way, the location of a geographic area.
The minimum bounding rectangle (xmin, ymin and xmax, ymax) defined by coordinate pairs of a data source. All coordinates for the data source fall within this boundary.
It should be noted that in the arcpy documentation for extent (though you haven't said which software you're using), the definition is essentially the same:
An extent is a rectangle specified by providing the coordinate of the lower left corner and the coordinate of the upper right corner in map units.
Extent Rectangle (you didn't ask for this one, but it was in the glossary so I threw it in):
[ESRI software] A rectangle that is displayed in one data frame, showing the size and position of another data frame.
The glossary doesn't have a definition for bounds, but I think it would likely be similar or identical to extent. It should be noted there is also a term called "metes and bounds", which is not really what you're asking, but I thought I would include it anyway.
A surveying method in which the limits of a parcel are identified as relative distances and bearings from landmarks. Metes and bounds surveying often resulted in irregularly shaped areas.
Finally, there is a tool in ArcGIS called Minimum Bounding Geometry which:
Creates a feature class containing polygons which represent a specified minimum bounding geometry enclosing each input feature or each group of input features.
Depending on which option you choose, the results will be different. However, it's worth noting that the Envelope option is a polygon which appears to be to be identical to the extent.