I would say every element should have a basic "staple" skill, likely a single target skill that does reasonable combat damage, and usable every combat, but beyond that, things diverge. Summarizing what has been stated and adding my own, you can diverge them as follow (maybe with auxiliary skills on top of mainline damage spells):
1. Damage level - high damage single target spells, low damage multi-target? how about critical hits and chances? how about damage randomness, where maybe wind (a "flighty" element) has a higher percentage of deviation compared to earth (a "stable" element)?
2. Targeting type - AoE, target all, single target only, maybe multi-hit, controllable targeting, random target (including friend or foe randomness)?
3. Skill cost - cost high MP, cost HP? maybe item cost? perhaps they all have warmups and cooldowns of varying degrees? how about limited use?
4. Environmental interactions - maybe fire can ignite certain environment? maybe wind can blow things away? maybe earth is ineffective against things floating or flying in the air?
5. Non-damage related effect - healing, turn manipulation, affects barriers, drains MP or HP, boost TP more, row/field manipulation, weather effect, etc? how about summoning, protection, or causes taunt or provocation? or perhaps some elements have buffs or debuffs or certain stats/params?
6. Status inflicted or interactions - maybe some element can inflict DoT, maybe some can inflict skill type sealing or even elemental sealing? maybe fire will inflict more damage on enemies afflicted with certain states, etc? maybe some states increase the effectiveness or improves the chances of an element's status effect? maybe the status in question is like a disease or in some ways spreadable?
7. Lore related - perhaps usable only by certain characters? if your characters have certain races or geography associated, only they can use those elements? in other words, the elements are strongly associated with certain geography or races? how about religion, if you world has it, and have elements associated to them and the pantheon? maybe some elements are the "primary" elements, and the rest comes as a result of combining them, and therefore have effect that mimics their "parents"? or maybe the elements are associated with colours? or maybe params?
One game I found which has interesting variations on the elements is Tales of Nebezem: Red Peril, where there are 6 elements (the 4 classical elements + Light and Darkness), and each element has a distinct specialty and/or effect (e.g. Light element has no effect on normal living things, and can do direct healing and revival, Water has regenerative effect and can cure statuses, Darkness has the single most damaging spell based on damage formula but cost HP to cast), a dedicated mage can have 2 elements (e.g. the healer has both Light and Water, a druid has both Earth and Water, etc), and each race has different elemental resistance.