Outside of context, you can't tell the difference between stone statue and a petrified creature by visual inspection.
The spell ends after 1 minute. RAW, there is no "lingering aura" in the case of flesh to stone as you originally asked. In many cases, finding a perfectly lifelike statue should prompt adventurers to ask questions: Why is this in the middle of a forest/underground cavern? Why does it appear to be terrified/running away from something? These aren't necessarily givens though. A medusa who throws fancy dinner parties would create completely different victims than a basilisk in an underdark prison complex. Also, art is just weird sometimes.
RAF, I don't think there is anything wrong with the DM ruling that the party can recognize that there is something magical about the statue, but I recently ran an adventure where statues of political dissidents were all over a city under martial law. The dawning horror of the party as they realized that these were all actually people over the course of several sessions was worth them not initially knowing that fact.
In the situation you described where a party needs to find a petrified prince in a room full of look-alike statues, the "perfectly lifelike" can be used to help the party. If they can discover errors in workmanship, differences in the material (perhaps the official statues are inlaid with metals or only carved from onyx), or have to solve some other logic puzzle, you can turn the guessing game into a memorable part of the rescue instead of a frustrating waste of spellslots.
As richarb points out, players can rely on non-visual characteristics though. A petrified creature is, by definition, a creature with the petrified condition. There are a number of divination spells that can locate creatures/objects, piece together the past, or otherwise provide insight into the situation, depending on your level of familiarity with the relevant cast of characters.