Abbreviated Model Descriptions
The Cox model is a survival model that cleverly models the hazard ratios through the observed ranks of the data, without needing to make an assumption of the underlying baseline distribution, but still requires the proportional hazards assumption.
The Tobit model is essentially standard linear regression, except that it can also handle censored data. The assumed distribution is then normal.
Pros and Cons
Pro: Don't need to make assumption about baseline distribution. This is very important for survival analysis: time-to-event data tends to be very not normal, often with extremely heavy right tails. Additionally, by only considering the rank of the data, you have a model that is more robust to the expected outliers.
Cons: Can be very difficult to interpret coefficient effects.
Pro: Simple extension of a model most analysts are already familiar with to allow for censoring, i.e. if all your data were observed and appropriate for linear regression (with one caveat mentioned in Cons section), then it would be appropriate to use a Tobit model.
Cons: Requires the assumption of linear effects and gaussian errors. In some applications, this is totally appropriate, but time-to-event data (i.e. survival analysis) rarely fits that criteria. Also, it's worth noting that the Tobit model is more sensitive to the normality assumption than vanilla linear regression.