It is basically an analytical type of difference. Both procedures are aimed at calculating the indigestible residue of a food substance, or, everything that is not fat, protein or carbohydrate.
Whereas the procedure for determining dietary fiber foresees the use of enzymes, raw fiber determination involves  chemical reagents only (acids and bases).
The procedure for determining dietary fiber exposes the sample to a series of enzymatic digestions that simulate the real digestive process which takes place in the human and animal digestive tract, calculating the undigested residue remaining at the end of the analysis.

On the other hand, in analyzing raw fiber the sample is digested using diluted solutions of acids and bases. Again the final undigested residue of the sample is measured. In this case the most widely used official procedure is the Weende method (official in Italy, France, England, Sweden and the USA).

Generally speaking, dietary fiber analysis is carried out on foods intended for human consumption whereas raw fiber analysis is carried out on animal feeds or on raw materials of vegetable origin, e.g. cereals.