The reason is that sulphur reacts with oils to form phytotoxic chemicals - and "phytotoxic" means "stuff that kills plants".
If you apply a spray containing oils (e.g. neem oil, or horticultural oils which are basically petroleum products) in enough quantity to do anything useful in killing pests, that counts as "using it".
Oil and water don't mix, so it would be safest to assume that rain doesn't affect the 30-day period. If the oil (or the sulphur) was washed off quickly by the rain, there would be no point in using it at all.
Sulphur is basically a fungicide. Oils are basically insecticides. Use whichever is appropriate to fix the problem you have.