They both determine the same thing, since in both these assays the assay signal is proportional to the number of viable cells (under the conditions of assay linearity). However, both these assays use different approach: neutral red assay (NRA) is based on the accumulation of neutral red dye in the lysosomes of viable cells, while MTT is based on the reduction of yellow tetrazolium MTT reagent by viable cells to purple formazan.
There is a paper reporting that the NRA is more sensitive than several other viability/cytotoxicity assays, including MTT (PMID: 18600217). I would say that NRA can be better than MTT when using cells with inherently low metabolic activity that would produce little formazan. The MTT and similar assays may be also inappropriate if cells are tested for viability upon treatment with inhibitors of certain enzymatic systems. On the other hand, the NRA is a heterogeneous assay and requires washing steps, while MTT, and especially its newer variants (e.g. WST), are homogeneous assays. However, all these assays measure absorbance as an assay signal, which is their limitation nowadays, when highly sensitive fluorescence (e.g. resazurin/AlamarBlue) and luminescence-based homogeneous viability/cytotoxicity assays are available.
Here is a link to an article comparing NRA, MTT and other cytotoxicity/viability assays:
Both assays determine cell viability. Nonetheless, their approach is different. Neutral red enters live cells and is retained in the lysosomes. Dead cells can’t retain the dye, that’s why the amount of retained dye is proportional to the number of viable cells. On the other hand, MTT measures the reduction of MTT to formazan crystals by metabolically active cells. Neutral red assay is more sensitive and usually presents less interference. Furthermore, MTT could be an unstable reagent.