Mythologically speaking, there are several forms of immortality.
Eternal youth: You age mentally but never age physically. You can die presumably of enough physical injury, possibly even disease.
Eternal life: You age physically and mentally, for all of time, you can still die the same as above.
Mundane Immortality: You age mentally but not physically, and you cannot die from disease(different versions still allow you to catch and carry disease, but they don't harm you), physical injury can still kill you.
Invincible Immortality: You age mentally but not physically, you are unaffected by disease and no wound can kill you, even if your heart was plucked from your chest.
These all generally apply to physical beings. Energy beings or ones that lack some form of organic nature(such as elemental) are generally considered to be Eternal, they can be physically dispersed, but never truly destroyed. Beings of the upper and lower planes often fall into this category as well, depending on their makeup and how they came to be. Devils are living beings that have been corrupted by evil, Demons are evil given physical form. The former can die, the latter can be physically destroyed, but it's evil essence often lingers or returns to the Plane of Shadow/Abyss to empower/form some other new creature.
If you were in my game and you were Immortal, I would agree that you gain the mental bonuses, but not the physical penalties of aging, but I would reduce them, or spread them out further. The immortal and incredibly long-lived often invest a long time in learning one specific thing, much more than "quicklings" like Humans do.