What are the main differences between the FreeSurfer volumetric approach and the statistical parametric mapping from SPM? In modern research of brain volume and function there are two main players. FreeSurfer which is free of cost and open source and the older, but more frequently used, statistical parametric mapping that itself is also opensourced but is typically used within Matlab, which is closed source.

What should I have in mind while deciding which one to pick for my research?
Freesurfer is great for mapping cortical thickness using anatomical MRI data, while SPM can be used to compare two populations, voxel-by-voxel. This is useful for PET imaging. For example you can see where in the brain amyloid uptake occurs in alzheimer's patients, vs healthy controls.

Our lab uses Freesurfer to process functional MRI data, so I don't have any experience using SPM for this. I will say though that probably the main difference is that many of the Freesurfer processes are simple via command line in UNIX, while SPM has a easy-to-use graphical interface.

SPM however also has command line functionality, so if you are comfortable with that, you might want to explore that. The bottom line is that you will just have to experiment with both programs to find what works for you. Methodologically, both tools should give you the same results, so it's really just a matter of preference, and depends on what you are studying.