Which is worse, plastic or styrofoam cups?
I suppose the answer depends on total lifecycle emissions as well as landfill environmental cost.
For plastic I assume PP or PETE. 
For styrofoam production is based on polystyrene so some styrene emissions.

For lifecycle costs, emissions for manufacturing as well as energy (coal or gas power plant) need to be considered. Also are either recycled and in what proportion and does that impact lifestyle costs. Not an easy question, or one I can quantify. For simplicity l would look at two questions- production emissions of the actual cup and or feedstocks and recyclability of each cup.  According to this link styrofoam is not recyclable so I would guess plastic is not as bad. 
https://sciencing.com/styrofoam-vs-plastic-cups-12400119.html
Here it says polystyrene is usually not recyclable locally. https://science.howstuffworks.com/environmental/green-science/polystyrene-recycling.htm
Here is a different perspective whose author suggests styrofoam lifecycle cost could be lower compared to some other options:
https://daily.jstor.org/is-the-30-year-long-styrofoam-war-nearing-its-end/

Not exactly an answer but at least a start