No, there's no difference. Detect magic would say if there was. Both types can detect the other's magic equally well.
Technically, not the even the different Knowledge skills apply to detecting magic itself. They apply to identifying the results of the two different kinds of magic, though, based on which one creates them most commonly - undead fall into Religion while constructs fall into Arcana. Note that this part isn't straightforward either; both kinds can summon outsiders (Know: Planes) and animals/magical beasts (Nature; which is associated with divine casters specifically through the Druid and Ranger).
For identifying actual spells, either as they are cast or by studying their ongoing effects with detect magic (and identifying magic items with either detect magic or identify), you use the Spellcraft skill. This doesn't change regardless of your or the other caster's traditions. Note that many spells that at first glance clearly belong on one side or the other quickly get mixed up when classes besides wizard and cleric get added to the mix.
Bards' spell lists read more like a cleric's with access to a few good wizard buffs. Cure spells, party buffs, and sonic damage like soundburst....
Domains, familiars, mysteries, and to a lesser extent bloodlines love to grant spells from the "other side". A Fire/Magic cleric has access to common wizard spells, for just a quick example.
No, detect magic doesn't automatically detect spellcasters as magical
The Sage Advice Compendium addresses a related question:
Is the breath weapon of a dragon magical?
If you cast antimagic field, don armor of invulnerability, or use another feature of the game that protects against magical or non-magical effects, you might ask yourself, “Will this protect me against a dragon’s breath?” The breath weapon of a typical dragon isn’t considered magical, so antimagic field won’t help you but armor of invulnerability will.
You might be thinking, “Dragons seem pretty magical to me.” And yes, they are extraordinary! Their description even says they’re magical. But our game makes a distinction between two types of magic:
- the background magic that is part of the D&D multiverse’s physics and
the physiology of many D&D creatures
- the concentrated magical energy that is contained in a magic item or
channeled to create a spell or other focused magical effect
In D&D, the first type of magic is part of nature. It is no more dispellable than the wind. A monster like a dragon exists because of that magic-enhanced nature. The second type of magic is what the rules are concerned about. When a rule refers to something being magical, it’s referring to that second type. Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
- Is it a magic item?
- Is it a spell? Or does it let you create the effects of a spell that’s
mentioned in its description?
- Is it a spell attack?
- Is it fueled by the use of spell slots?
- Does its description say it’s magical?
If your answer to any of those questions is yes, the feature is magical.
Let’s look at a white dragon’s Cold Breath and ask ourselves those questions. First, Cold Breath isn’t a magic item. Second, its description mentions no spell. Third, it’s not a spell attack. Fourth, the word “magical” appears nowhere in its description. Our conclusion: Cold Breath is not considered a magical game effect, even though we know that dragons are amazing, supernatural beings.
Detect magic, like other game mechanics, operates by this same logic with regard to what is considered magical. The spellcasting abilities of creatures (innate or otherwise) are considered "the background magic that is part of [...] the physiology of many D&D creatures". Detect magic is designed to detect magical effects, not the background magic that suffuses creatures or the universe.
Chris Perkins confirms this sort of interpretation here:
Can detect magic detect magic potential of spellcasters even if they're not actively casting a spell?
It's not a wizard detector, if that's what you mean.
Given that the question he's responding to asks about spellcasters in general, it seems clear that his response is not specific to wizards - he's suggesting that the spell doesn't automatically detect spellcasters simply due to their magical abilities.
Best Answer
The duration on Detect Magic is a bit more detailed then that:
In order to use it, you have to keep concentrating. The more rounds you concentrate on a given target area or subject, you get more information as detailed in the spell.
You can turn to another target area or subject, and so long as you keep concentrating the spell lasts the full duration. But the information you get on the new target will depend on how many rounds you concentrate on it.
At level 1 you have 10 rounds maximum, so you could do the full 3 round duration on a sword, then 3 more on a shield, then 3 more on Bob the Barbarian, then 1 round on something else (with just one round looking at something, you get less information, as described in the spell).