A Depth Sounder or fathometer, both being a specialized type of echo sounding system, a type of Active SONAR. ('Sounding' is the measurement of water depth, a historical nautical term of very long usage.) The Depth Sounder uses active sonar to detect fish and 'the bottom' and displays them on a graphical display device, generally a LCD or CRT screen. In contrast, the modern fathometer (from fathom plus meter, as in 'to measure') is designed specifically to show depth, so may use only a digital display (useless for fish finding) instead of a graphical display, and frequently will have some means of making a permanent recording of soundings (which are merely shown and subsequently electronically discarded in common sporting fishfinder technology) and are always principally instruments of navigation and safety.
The distinction is in their main purpose and hence in the features given the system. Both work the same way, and use similar frequencies, and, display type permitting, both can show fish and the bottom. Thus today, both have merged, especially with the advent of computer interfaced multipurpose fishfinders combining GPS technology, digital chart-plotting, perhaps RADAR and electronic compass displays in the same affordable sporting unit.