Is Thiessen polygons the same thing as Voronoi polygons? I am using ArcMap 10 and also QGIS 2.4 and I would like please to know the exact difference (If any) between the two methods.
Yes, they are the same thing. In the field of GIS we tend to refer to them as Thiessen polygons, after the American meteorologist who frequented their use. In other fields, particularly mathematics and computer science, they are generally referred to as Voronoi diagrams, in honour of the mathematician Georgy Voronyi. Both uses are acceptable.
We cannot know the exact difference because we cannot see the source code of ESRI's implementation. However, it appears from a cursory glance that the two implementations do, in fact, utilize the same method that is a rough translation of Steven Fortune's classic sweepline algorithm.
Here you can have a look at the actual source code that is used in QGIS. It includes the following description:
For programmatic use two functions are available: computeVoronoiDiagram(points) Takes a list of point objects (which must have x and y fields). Returns a 3-tuple of: (1) a list of 2-tuples, which are the x,y coordinates of the Voronoi diagram vertices (2) a list of 3-tuples (a,b,c) which are the equations of the lines in the Voronoi diagram: a*x + b*y = c (3) a list of 3-tuples, (l, v1, v2) representing edges of the Voronoi diagram. l is the index of the line, v1 and v2 are the indices of the vetices at the end of the edge. If v1 or v2 is -1, the line extends to infinity. computeDelaunayTriangulation(points): Takes a list of point objects (which must have x and y fields). Returns a list of 3-tuples: the indices of the points that form a Delaunay triangle.
Now we can't see ESRI's proprietary code that drives their tool, but their documentation's description immediately reveals that the basis behind both tools is the same:
Thiessen proximal polygons are constructed as follows:
All points are triangulated into a triangulated irregular network (TIN) that meets the Delaunay criterion. The perpendicular bisectors for each triangle edge are generated, forming the edges of the Thiessen polygons. The location at which the bisectors intersect determine the locations of the Thiessen polygon vertices.
The actual nuances of the code driving the two are obviously different, as it has been demonstrated that Bill Simon's translation has known bugs that are not present in ESRI's version.
There are (as has been stated in comments above) several other different ways to generate Voronoi diagrams, even in GIS, such as this raster based methodology. There are also other vector based methods to generate Voronoi diagrams in GIS.
There are several advantages and disadvantages to each of the methods. For example, Fortune's algorithm is relatively fast and well documented, but currently there is no known way to generate multiplicatively weighted Voronoi diagrams using his direct implementation.
Raster methods are generally much slower computationally but allow for creation of different types of Voronoi diagrams (such as farthest point Voronoi diagrams) without completely reinventing methodology.
Full disclosure: I have worked as a research assistant to the professor who wrote the paper for the raster based methodology for generating Voronoi Diagrams.
TL;DR: Though the actual implementations differ slightly, they are based upon the same algorithm and both should produce the same result (aside from the few edge cases that produce the bugs noted in Dan Patterson's question linked above).