Untreated acid reflux is a progressive disease. Early stages characterized by episodes of heartburn are not associated with any signs of damage. later on signs of inflammation of the esophagus (esophagitis) are evident.
Barrett's esophagus is a warning sign because it is considered a precancrous stage. To understand Barrett we should recall the normal cellular lining of the esophageal lumen which is a specialized type with characteristic number, shape and cell arrangement. This particular type is partly replaced by a cellular pattern characteristic of intestinal lining membrane. Such a cellular modification is a sort of adaptation to damage induced by acid reflux.
Later on acid reflux disease may be complicated by malignant transformation or cancer on top of Barrett's esophagus called Adenocarcinoma.
To summarize:
Acid reflux: only symptoms(e.g heart burn).
Acid reflux disease: evidence of esophageal damage.
Complicated reflux disease: Barret's esophagus which may result in cancer.