An OCO is a contingency type of order, contingent on the first part of the order to get filled before the second part gets cancelled. The other bracket orders mentioned are not contingent on anything. You can use all kinds of bracket orders to either enter or exit the market. To play breakouts you can bracket market consolidation with buy stop and sell stop or if you want to pick tops and bottoms you can bracket market with limit orders. To exit trades you can bracket with limit and a stop, etc

I've been trying to find the difference between an OCO order and a bracket order.

My understanding is OCO is essentially a bracket order.

So when would use a bracket order versus a OCO order?

For a typical trade with a take-profit and a stop-loss, it is described like this:

BRACKET (
....Entry Limit Order
....OCO [
........Take-Profit Limit Order
........Stop-Loss Market Order
....]
)

The bracket means the Take-Profit and Stop do not become active until the entry order is filled.
The OCO means if the Take-Profit is hit, the stop is cancelled, or if the stop is hit the take-profit is cancelled.

It is also possible to combine multiple brackets and OCO orders. For example, if price is trading in a range and you want to enter when the range breaks, but you don't know whether it is going to break bull or bear, you can setup OCO bracket orders for Stop entries both top and bottom, and that will get you in the market whichever side breaks first, and will also manage the appropriate take-profit and stop orders for each side.

OCO [
....BRACKET (
........Bull Stop Entry
........OCO [
............Take-Profit Limit Order Bull side
............Stop-Loss Market Order Bull side
........]
....)
....BRACKET (
........Bear Stop Entry
........OCO [
............Take-Profit Limit Order Bear side
............Stop-Loss Market Order Bear side
........]
....)
]