There are many points to note when using the causative. If you want the simple rule,
transitive verbs only take に, while intransitive verbs take に only if there is を already, if not use を. Many native speakers just simply avoid putting に twice even if it's perfectly fine (in some situations it's wrong and others it's disliked), and when they work that logic in reverse, they believe に is wrong if を doesn't exist...
For the detailed rule, we'll need examples, and a cup of coffee...
A transitive verb, so you must use に.
An intransitive verb, both are correct, The difference between the two is を is permissive while に is compulsive. You can notice the difference in the negated sentences, "子供を遊ばせない" would translate to "won't let kids play" while "子供に遊ばせない" would be "won't tell kids to play" or "won't make kids play".
Naturally, you'd find native speakers mostly use を and might even say に is wrong.
When there's a 結果的に nuance, only を is usable.
What about emotion verbs?
Emotions aren't controlled by the will so I can't force my will on my friend to make him cry and hence can't take に.
The winter can't force its will on the sky to make it snow because the winter is inanimate and causative doesn't like inanimate causers. So instead of being able to use both we can only use を here.
Had it been an animate causer,
Unlike 食べる which is transitive by nature, 落とす is trully a causative. We'll treat this is as a normal non-causative conjugation of a transitive verb so we'll use を. The reasoning is that the verb it conjugated from 落ちる(落つ) is a 非意志動詞, same as 降る.
We've seen verbs which take を for their objects, what about に verbs?
the simple rule? doubling に is bad so use を. Actually, に is reserved for a specific usage with these verbs and don't like doubling in this case. If it didn't have に in the original sentence before the causative you could've used it just fine. and the difference would be similar to 子供（を・に）遊ばせる. And you'll find natives saying に is wrong in that case because intransitive take を but they are incorrect.
また②は、 私が行こうとしているとき、誰かが行かせまいとしているような状況で、私が行くのを止めないでください、という表現。 にも使えます。
Courtesy of 快仁21面相さん on this chiebukuro question.
What about verbs that take を for locations?
Can't use を twice, hence に. I haven't seen anything concrete for if を would be acceptable If you remove トラックを or just plain wrong. But ※I know natives persist that に is wrong when they don't see トラックを. So I assume both are acceptable and the difference is allowance vs compulsion?
※You'll find foreigners believing に is allowance while を is compulsion in that link, I personally found sources made for teaching foreigners which say this. It's plain wrong.
For the last one, a bonus!
Verbs where you do stuff towards people like speaking take に. Luckily most native speakers agree on this and don't use を. consider this the special rule to the simple rule.
Some research by Kansai Gaidai University (関西外国語大学) I used as reference for examples and what people prefer in different situations. (pdf)
Well that took a couple of hours... and I didn't make coffee...