Q:

Using Proportional Symbol Map vs Graduated Symbol Map?

I am having trouble understanding the difference between a Proportional Symbol Map and a Graduated Symbol Map, or why it matters. 
How do these two differ from each other?

A:

The main difference between the two is that graduated symbols are a "classed" symbology while proportional symbols are "unclassed."  While most cartographers use the terms “proportional point symbol map” and “graduated point symbol map” interchangeably, in ArcMap these two terms have specific meaning. In the software, proportional symbol maps use absolute scaling or apparent magnitude scaling and graduated symbol maps use range grading. With proportional symbols, ArcMap allows you to set only the size of the smallest symbol (from which other symbols are scaled upwards). You can also use Flannery Compensation to adjust the symbol sizes.
Quoted from source below:
When to Use
Proportional symbol maps scale the size of simple symbols (usually a circle or square) proportionally to the data value found at that location. They are a simple concept to grasp: The larger the symbol, the "more" of something exists at a location. The default setting in indiemapper is to scale the circles directly proportionate to the data (the "unclassed" tab) so that if, for example, Toronto has twice the population of Vancouver, the population symbol for Toronto will have twice the area. However, in indiemapper you can also group your observations into categories or numerical ranges (the "classed" tab) and created graduated symbol maps that may, for example, only have three symbol sizes corresponding to three categories of city size (e.g., cities of <1 million, 1-4 million, and over 4 million people). The pros and cons of proportional versus graduated symbols are discussed in more detail below.
See data classification for a more general discussion (should you chose to go down that road here).
One further note: With 2D proportional symbols like circles and squares (see example below), it is the area of the symbols that encodes the data, not their height or length.
Reasons Why We Like Them
Proportional symbol maps are very flexible because you can use either numerical data (e.g., income, age) or ordered categorical data (e.g., low, medium, and high risk of bankruptcy). They're also flexible because they can be used for data attached to geographic points (e.g., a precise location) or data attached to geographic areas (e.g., countries).
One advantage of proportional symbol maps over dot density maps is it is generally easier for map readers to extract numbers from the map since estimating the size of a symbol is less tedious than counting many little dots. An advantage of proportional symbol maps over choropleth maps is that the size of the enumeration unit doesn't matter: If a country with a small geographic area, such as the Netherlands, has a large data value attached to it, it will have a large symbol over it. By comparison, on a choropleth map, smaller places are easily overlooked on a busy map-even if they have large data values-while large countries such as Canada dominate the map no matter what color they are. It can be argued, thus, that proportional symbol maps "let the thematic data speak for itself," since the size of the symbols relates directly to the thematic data and not just the footprint of the enumeration unit. Lastly, unlike choropleth maps, proportional symbol maps can use either raw data (totals, counts) or standardized data (percentages, rates, ratios); choropleth maps should only be made with standardized data.
source: indiemapper
other references:
http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?id=292&pid=286&topicname=About_symbolizing_data_to_represent_quantity
https://www.mtholyoke.edu/courses/tmillett/course/geog205/files/Symbolizing_Data.pdf
http://webhelp.esri.com/arcgisdesktop/9.3/index.cfm?id=292&pid=286&topicname=About_symbolizing_data_to_represent_quantity

