5e: Direct information on climate is limited and scattered
Currently the Sword Coast Adventurer's Guide is the best source of 5e information on the sword coast. Unfortunately (for our purpose anyway) its focus is on political groups and mentions of the climate in a given area are rare. The island realm of Evermeet is mentioned as being, "the island
paradise of the elves", and since most of its population is sun elves perhaps we can assume it's pretty nice? We also find out The Cold Lands are , "...in the cold, dry environs of the northeast", which is great but sadly not actually on the sword coast.
There are a couple of maps in the SCAG that we can use to infer about climate. The map of the coast shows areas near the sea of swords to be (mostly) grasslands, a couple of marshes as you move further out, and then the desert of Anauroch even further out. From this we could conclude that a fair bit of rain comes in off the sea and fades out before getting too far inland. The second map is of the north and shows that it is, shockingly, cold.
Alright well how about Hoard of the Dragon Queen. The adventure takes us north along the sword coast so we can look for references to the climate in the areas we pass through. The first mention of bad weather is around the Mere of Dead Men is, "...cold, muddy...", which is great news! The Mere of Dead Men is located north of waterdeed, right on the coast, and hemed in by mountains. Its waterlogged status fits our theory that rain from the sea is a dominant force (being held back by the mountains the rain would be very heavy in this area). Also since up to this point it has been nice enough to mention it being cold we can assume that south of here it's at least mild.
3e: There's a whole book on it
If you're interested in information from other editions then you are in luck! The Forgotten Realms Campaign Setting from 3rd edition has reams of geographic and climate info. A sidebar on page 78 about Waterdeep's climate info discusses its latitude, the nearby oceans moderating aspects on its weather, and the annual arrival of arctic winds. The level of detail available is great and you can easily find information the climate of any specific area you're interested in.
Time of Casting/Acting is Important
If a Druid were to attempt predict the Weather before the spell Control Weather was cast, then they would have no idea that the weather was changing because they would not have that information.
If they were to attempt to predict the Weather after the spell was cast, then I would rule that they certainly did have that information.
Also, the pertinent Druidcraft text reads:
You create a tiny, harmless sensory effect that predicts what the weather will be at your location for the next 24 hours. The effect might manifest as a golden orb for clear skies, a cloud for rain, falling snowflakes for snow, and so on. This effect persists for 1 round.
Thus, since it only lasts 1 round (or 6 seconds) it can't predict any use of Control Weather after the spell, because that would be like saying Detect Magic could be used to detect magic through prediction after its duration had ended. The effect has a Duration: It last for 6 seconds and then disappears, and cannot be retroactively used to predict the use of Control Weather.
EDIT: It's been how many months and I forgot to include this simple Caveat? I just want to throw in that you would have no indication other than common sense whether weather effects were magically created or not (for example, if you see an intense blizzard in a desert that hasn't gotten any precipitation for the last 250 years, something might be up.)
Best Answer
It's very reliable. According to the description, it "predicts what the weather will be". No ifs, ands, or buts. This suggests perfect accuracy.
That said, it's not very precise. (Precision is not the same as accuracy.)
The description isn't clear whether it shows a single effect that predicts the weather over 24 hours, or if it changes in appearance over the course of the round (6 seconds) depending on the weather. Either way, if it shows you that you have a cloudy night, that doesn't mean it will be cloudy for the entire night, or at any particular moment. So if you're sneaking into a castle, be prepared for the moon to shine through an inconvenient gap in the clouds.
Similarly, it says it shows the weather "at your location". This is equally imprecise - it could be anything from the exact point you're standing at the time to the entire region you happen to be in. In the first case, unless you snuck into the castle to do your weather prediction for sneaking into the castle, be prepared for the weather to be a little bit different. In the second case, you only have a very vague idea. The city where I live often predicts rain, but I don't often see any - it mostly rains down the south end.