Re:
The main thing that affect grip level for both car and bicycle tyres is the rubber compound. The second thing that affects grip is tread pattern and this interacts with compounds in a few ways.
So first off there are two ways tyres produce grips. One being chemical grip and the other mechanical grip.
Chemical grip is affected by the compound and produces the bulk of the grip from contact between tyre and road. More contact area = more grip which is why racing car/bike tyres are slicks (except when it rains for reasons explained below).
Mechanical grip is to do with how the tyre profile interacts with imperfect road surfaces (which let's face it they all are unless you're in a velodrome). A big tread will purchase around uneven surfaces, but at the expense of contact area. As this is smaller than chemical grip, slicks are still a better choice on road. Off road this is reversed.
As had already been mentioned, another factor is aquaplaning. In a larger contact area, the tyre can float on water losing grip altogether. However this isn't an issue for bicycle tyres because they're so narrow. If you think about it, the tread blocks in your car tyres are probably wider than your bike tyres!
Other considerations for heavy treads is they make tyres more puncture resistant and if you need to ride in snow, a big tread will pick up snow and snow sticks to snow better than rubber.
Temperature is also a factor for both kinds of grip. In cold weather soft compounds move better than hard, so give more mechanical grip.
So there you have it, get slicks, unless you're riding poor surfaces, or in snow, or deep puddles or cold weather (probably), in which case still get slicks
