HREW tube is Hot Rolled, electrically welded tubing. Usually mild steel, although it often comes in decent material like 1018.
DOM, starts out exactly like HREW, then is drawn over the mandrel to tighten up the tolerances. Usually same grade or better than HREW. If the sizes are equal the strength differences aren't that astounding IMO. Yes, DOM is also welded seam tube...the welded seam is just a lot harder to see.
All tube is defined by the outside diameter. Example: 1.75" tube is 1.75" outside. The thickness of tubing is given as "Wall thickness" EG: .120" wall
All pipe is defined by the inside diameter and even that is pretty loosely adhered to. 1.75" pipe should be near to 1.75" inside diameter. The thickness of pipe is given as 'schedule'. EG sched 40. Many steel catalogues will have a thickness table telling you the exact wall thickness for any given pipe size and schedule.
The problem with pipe is there are a lot of grades of pipe, and you could inadvertantly get something not well suited to cage or door construction.
Rene