We present the results of a 4 ks ROSAT PSPC observation pointed toward the NE region of the Vela supernova remnant (SNR). The image counting statistics allowed us to perform, for the first time, a spatially resolved spectral analysis of the X-ray emission on an angular scale of 5' (similar to 0.5 pc assuming a distance D = 350 pc). Accounting for systematic uncertainties in the PSPC energy response, a simple one-temperature Raymond-Smith model yields an acceptable fit to the data in 69% of the spatial bins considered (out of 273), at the 95% confidence level. Using this model we discovered statistically significant variations of temperature (0.10-0.28 keV) and interstellar hydrogen column density (log N-H = 19.5-20.5 cm(-2)) on the spatial scale explored. We interpret this result in terms of SNR shell expansion in a nonuniform medium. On the other hand, the data as a whole are locally best fitted by a two-temperature Raymond-Smith model, with T-1 ranging between 0.05 and 0.22 keV, T-2 in the range 0.2-0.9 keV, and with a ratio of emission measures EM(2)/EM(1) less than or equal to 0.5. This result improves the previous single-temperature model description of the Vela SNR emission, based on lower spectral and spatial resolution Einstein data. We discuss the physical interpretation of the two-temperature model, suggesting that it can be explained by the under-ionization of the shocked plasma, as predicted by Itoh.