Is it good idea to max skills? If it is then does it increase it's stats if you continue putting after unlocking all skill cubes in the skill or does it has other benefits that I don't know?
Skill stats continue to increase as you add points (check skill description).
Other benefits won't unlock, after skill cubes and/or purification equipment modifying skill, that's it.
My personal suggestion is to invest any spare SP after capping special moves and desired passives into a commonly used active used for DPS, or an active that provides stronger buff/debuff with level increase.