Fucoidan, a fucose-containing sulfated polysaccharide with diverse biological functions, is mainly recovered from brown algae. In this study, we utilized a compressional-puffing process (CPP) to pretreat Sargassum crassifolium (SC) and extracted fucoidans from SC by warm water. Three fucoidan extracts (SC1: puffing at 0 kg/cm2; SC2: puffing at 1.7 kg/cm2; and SC3: puffing at 6.3 kg/cm2) were obtained, and their composition, and antioxidant and neuroprotective activities were examined. The results suggest that CPP decreased the bulk density of algal samples, expanded the algal cellular structures, and eliminated the unpleasant algal odor. The extraction yields of fucoidans were increased and impurities of fucoidans were decreased by increasing the pressures used in CPP. The SC1–SC3 extracts displayed various characteristics of fucoidan as illustrated by the analyses of composition, Fourier transform infrared (FTIR) spectroscopy, and molecular weight. All three extracts SC1–SC3 showed antioxidant activity dose-dependently. Although both SC1 and SC2 possessed high and similar neuronal protective properties, SC2 showed a higher extraction yield, higher efficacy in the reversion of H2O2-induced cytotoxicity in rat pheochromocytoma PC-12 cells, and lower impurities compared with SC1, and thus SC2 is suggested as a good candidate for a therapeutic agent in the preventive treatment of neurodegenerative diseases.