Do permanent spells like flesh to stone leave a lingering aura so you can tell a statue from someone who has been petrified? Or do you need to guess if that statue of the prince is actually the prince himself?
The flesh to stone spell has instantaneous duration. Thus there is no “normal” aura after the spell is cast. But it is generally agreed (I can't find unambiguous RAW entry for that) that since magic is present at the time of casting, the spell leaves lingering aura as normal. The flesh to stone spell is level 6. It results in moderate aura at the time of casting and a lingering aura persisting for 1d6 minutes (see detect magic). After that no aura can be detected.
Your option after the time has passed is the true seeing spell.
The subject sees through normal and magical darkness, notices secret doors hidden by magic, sees the exact locations of creatures or objects under blur or displacement effects, sees invisible creatures or objects normally, sees through illusions, and sees the true form of polymorphed, changed, or transmuted things.
Other similar spells or spells specifically designed to detect creatures turned to stone may exist in other sources that I don't know. Alternatively you can work on researching such a spell with your DM if he/she allows spell research.
At the same time spells are not the only way to tell petrified creature from a statue.
In D&D 3.5 Spellcraft skill can be used to identify materials made by spells.
20 + spell level: Identify materials created or shaped by magic, such as noting that an iron wall is the result of a wall of iron spell. No action required. No retry.
Pathfinder has this option for Knowledge (Arcana) skill:
Identify materials manufactured by magic Arcana: 20 + spell level
Finally, you can undertake a regular investigation.
For example, you can learn what clothes the prince was wearing before disappearing, or you can find out what statues of said prince were ever made (if any) and where they are situated. Based on this knowledge you can make a decision on whether this statue is a regular statue or a petrified prince.
People in the comments also note that a statue will bear marks of the instruments used to sculpt it, thus a Profession (Sculptor) check may help to discover them.
I would think that a petrified creature would be an extraordinarily detailed "statue". It would immediately stand out as unusual.
Only the very greatest of stone-carving artisans would be able to come close to the precision and fine detail.
In an area that even only rarely had a basilisk or cockatrice visit, any highly detailed stone creature would likely first be assumed to be turned to stone rather than carved. Especially if it is in a cave or on a game trail in the woods. In fact, unless the setting and pose of the statue was sufficiently "statue-like", assuming it to be a creature turned to stone would be natural.
In fact, any highly-skilled stone-carver may actually be suspected of turning things to stone rather than carving them. I think there was an old Gummi Bears cartoon with this plot point.