What are the main differences between a principal component analysis and a factor analysis?

Here is a graphic I created to answer this. Note that the assumption is that the rectangular box contains the total variance of a model. The columns in the first figure are the features (variables) in the model. In the second figure, each colored section is a PC.

They’re built for different things. I assume your question is related to psychology so I’ll try to answer appropriately.PCA will always net you better results. Suppose you’re making some scale and the items measure the constructs (or factors) you’re interested in. When you get the results for factor loading, the numbers will always be bigger in PCA. This means that the items will always “fit” the factors better, it will be easier to deduce the items 3, 4, 6, 5, and 10 describe factor X. Factor analysis will often give less pink results, which is why some people (intentionally) will prefer PCA over Principal Axis Factoring just to commit fraud.A better technical answer is provided here: http://stats.stackexchange.com/questions/1576/what-are-the-differences-between-factor-analysis-and-principal-component-analysiIf you want to understand what the model differences are, you can always look up the Wikipedia article on it.And, if you want to know how to avoid misusing factor analysis, read and follow this: https://scholar.google.bg/scholar?q=in+search+of+underlying+dimensions&hl=en&as_sdt=0&as_vis=1&oi=scholart&sa=X&ved=0ahUKEwju46HBgdfNAhXC3SwKHS_PCP0QgQMIGjAA(it’s the first article)

The most important difference is that there is no residual in PCA. It is in this sense that PCA is not a “model” and delivers a mathematically unique solution. This is not true for FA which does posit a residual, is a model and, once the solution is rotated to a Thurstonian “simple structure,” is no longer mathematically unique.