Find Familiar can be useless or overpowered
It heavily depends on how the DM treats the creature. Is it a second player's character, a reliable magical device, or a living pet (extremely loyal, but still independent)? Many tables stick to the former option as easiest one, despite the rules suggesting the latter. Which is not a bad thing, but make sure you're on the same page with the DM before choosing the spell.
Here's a bunch of possible downsides of the Find Familiar spell comparing to the Clairvoyance:
The familiar might have trouble with getting to the spot
The Find Familiar spell says the creature "always obeys your commands". However, it always acts independently. The results may vary, since every creature is definitely limited by its own physical and mental capabilities. Even staying in the same place can be challenging (you can try this yourself).
So the Wizard just says to a rat "go to the Chancellor's Room and wait there" and hopes it understood the task and can handle it. Fortunately, a rat is not a spider — the latter has the lowest INT score possible and worse navigation skills.
Probably not a problem if the Wizard can physically visit the room beforehand and just leave her/his familiar there. But for how long? Let's find out.
The familiar needs supplies to sustain itself
The familiar is a living being, and nothing in the rules says it doesn't need food, water or sleep. See Do familiars cast from Find Familiar need to eat and sleep?
You can't just plant it in a room and leave it there for days. A bird cage with food it probably the best option for a (relatively) long-term surveillance. On the other hand, spiders can easily survive without food for days. However, here's another problem...
Familiar's senses might be limited
Spiders have eight eyes but poor eyesight, so they rely on touch and vibration instead. The same problem is true for bats, rats, lizards and most other forms except for birds and cats. The spell allow you to "see through your familiar's eyes", but it doesn't improve the creature's eyesight. The same is true for hearing — spiders are virtually deaf: they perceive sounds as touch. The DM might rationalize, that the spider isn't capable of conventional hearing, so when you "hear what it hears", you hear nothing.
And if a spider form was chosen as it's the smallest one, there's might be a problem with that.
A Tiny Spider is not so tiny
When you cast Find Familiar, the creature takes an animal form you choose from the list. However, you don't choose the particular size. You can't demand a extraordinary small cat. It is just a Tiny Cat. Or a Tiny Spider. Which can be the same size as the cat. This would be reasonable, since the familiar has to be big enough providing its capabilities of helping in combat or carrying up to 15lbs. See What, approximately, is the size of a Tiny creature?
The familiar is vulnerable
An obvious thing, but worth mentioning. Even if you managed to plant a little inconspicuous spider in the target room, and the DM ensures the spider gives you a perfect vision, the success is still not guaranteed. The spider can be eaten (by another spider) or smashed (deliberately or by accident). Comparing to this gambling, Clairvoyance is fail-proof and always leads to expected results.