Every class proficient with martial weapons is also proficient with simple weapons, so there would be no mechanical advantage to using a trident over a spear (unless in some unforeseen circumstance you needed the extra 1lb).1
There would be a reason to use a spear (simple weapon) and not a trident (martial weapon), but not the other way around. Bards, clerics, druids, monks, rogues, and warlocks are proficient with simple weapons but not martial weapons. For them, their proficiency bonus would be an advantage of the spear over the trident.
However, these are not the only weapons with the same mechanics (other than price and weight). The following martial weapons are mechanically identical:
- battleaxes and longswords are both versatile weapons causing 1d8 (1d10 two-handed) slashing damage
- morningstars and war picks both cause 1d8 piercing damage
- The “Variant Entertainer: Gladiator” background (PHB, p. 131) allows a player to “replace the musical instrument in your equipment package with an inexpensive but unusual weapon, such as a trident or net.” RAW this only gives the trident as equipment. However, a DM could house rule that it also grants proficiency (in place of the music instrument from the Entertainer background). If allowed and taken by a sorcerer or wizard, they would have proficiency with tridents but not simple weapons (i.e, spears).