High spatial and spectral resolution spectra of the near-infrared H-2 emission from the planetary nebula BD +30 degrees 3639 are presented. A 7 " x 13 " portion of the nebula was mapped at a resolution of 6000 in the upsilon = 1-0 S(1) emission line of molecular hydrogen, revealing the morphology and kinematics of the H-2 emission, These spectra show large velocity shifts in the H-2 line across the nebula. Comparison with single-dish CO observations show significant velocity differences between the peaks of H-2 emission and CO emission. Such differences are unexpected for a photodissociation region (PDR), since PDRs are supposed to form in the surface of the cold gas las is the case for NGC 7027). A single long-slit spectrum which includes the peak of the H-2 emission and covers the entire K band at a resolution of 1000 is analyzed. Fourteen molecular hydrogen emission lines are detected in a 2 " x 0." 5 region covering the bright northeastern H-2 lobe. The line ratios indicate that the H-2 emission is radiatively excited, but that the gas density is large enough (n greater than or equal to 10(4.5)-10(5) cm(-3)) for collisions to thermalize the lowest vibrational states. The measured ratio of ortho-hydrogen to para-hydrogen is 2.51 +/- 0.22, slightly smaller than the equilibrium value of 3. Spatial variations of the upsilon = 2-1 vibrational temperature and the upsilon = 1 ortho/para ratio are consistent with a decrease in the relative proportion of collisional excitation as distance from the central star increases.