It is true that both of these processes involve conversion to ammonia/ ammonium but the starting nitrogen form is different. Ammonification is the mineralization of organic nitrogen into ammonia. Dissimilatory nitrate/nitrite reduction to ammonia (aka DNRA) starts with inorganic nitrogen (nitrate/nitrite), not organic N. The nitrate/ nitrite is then reduced to ammonia/ ammonium.
Just to add a small part to the answer: DNRA is believed to occur under highly anoxic conditions, such as anaerobic waste reactors, anoxic sediments, and bovine rumen, while ammonification occur as part of the mineralization process when heterotrophic soil organisms consume organic matter for energy and C.