I will answer the easiest first: “What are the pros and cons of each technique?”
The general answer to this is that T1 scans are better for anatomy, and T2 are better for pathology, since T2 highlights fluid and many pathologies (including almost all traumatic or neoplasmic brain pathologies) are accompanied by oedema. However this rule is not always correct, because gadolinium contrast can only be imaged by T1, not T2.
“What actually happens to the particles in the brain when each one is performed?”
MRI physics is insanely hard, but I will do my best to summarise it for you. An MRI scanner works by applying a strong magnetic field to the body, which aligns all the protons along a single axis. A radio frequency (RF) pulse is then emitted, which knocks the protons off this axis. The magnetic field causes them to return to alignment, and in the process they emit RF waves which are then detected to form the image.
“What are the methodologically differences in acquiring them both?”
There are two types of relaxation; longitudinal and transverse. T1 imaging uses longitudinal relaxation signals to form its image, and T2 uses transverse relaxation signals.
When the protons are in equilibrium, there is a small discrepancy between spin ‘up’ and spin ‘down’ protons (this is quantum spin). If a 90 degree RF pulse is sent out, we can change this so there is no discrepancy, and there are equal numbers of spin up and spin down. When the pulse is turned off, the discrepancy returns. The signal from the few protons that switch back forms a T1 image.
The main magnetic field causes all the protons in the field to be aligned, but they retain their individual precession frequency. When an RF pulse is applied, they begin to precess in phase with each other, and a signal can be detected. When the pulse is switched off, they slowly return to their previous precession frequency, which is dictated by their binding in their respective atoms. As they lose phase the signal drops, and the time taken for the signal to disappear is measured and used to form a T2 image.