Diesel has a high content of aromatic components and a relatively low flash point. Mineral oils (distilled from petroleum crude) tend to have a much lower aromatic content and higher flash point. There are a wide range of oils (low tox, paraffins, olefins, esters, etc.) in addition to Diesel and Mineral oil.
The aromatic content of Diesel makes it more compatible with a lot of the additives used in formulating a drilling fluid, however there are disadvantages from a HSE perspective due to the toxicity of the aromatic components and the risk of combustion. Mineral oils generally have less toxicity and less combustion risk due to the higher flash point. The cleaner nature of these oils can make them less compatible with some of the simpler chemical additives in a drilling fluid.
Biodegradability is a harder question to answer as it will depend on the environment and the particular Mineral oil. Also, the volatile components in diesel can be evaporated in a hot environment. Some of the other oil types have much better biodegradability.