This question got me thinking: any question tagged spells can also be tagged magic, and they seem to serve a similar purpose. What's the difference between the magic and spells tags, or are they similar enough for one to be synonymed into the other, or for them to be replaced with a new tag?
Speaking from a D&D point of view, the two things are pretty different. Spells is only one of the possible manifestations of magic along with spell-like abilities (most "spellcaster" monsters and prestige classes), supernatural abilities (most other sort of magical monsters and classes), invocations (dragonfire adepts and warlocks), powers (psions, but it might as well be a completely different thing than magic under different optional rules), mysteries (shadow magic), enchantments (items) and magic effects not replicable by anything in the rules.
So, spells is specific and magic is generic.
I'm perfectly aware this only apply to D&D 3.x, previous D&D editions and Next (D&D 4e has a different concept of spells that is more fluff-related but still has spells as a subset of magic), but a distinction it is.
In the question you linked, Detct Magic is a spell that can detect any form of magic.
Quite frankly it's sometimes a semantic issue. Magic is not always an active thing. In L5R there's something called Kiho which are magical effects, but ultimately these are not spells. There abilities that are channeled through things like Shadowlands Taint and The Nothing that are magical in nature (as in supernatural) but are not spells. What does count as spell use is the traditional magic forms used by shugenja that quite literally derive from written scrolls, spoken and sometimes performed in a specific fashion so that there is any result at all let alone the desired effect.
In essence, magic can exist in many passive forms, sometimes even active ones but a spell almost always between the systems I have personal experience with relates to something that requires exact and almost ritualistic execution.