The mechanism of copper ion toxic to cell including at least two reactions.  One is copper ion will bind thio group in protein, the other is copper ion catalystic reactive oxygen species ROS product in sequence reaction processes.

"Why is copper is toxic to the cells?": copper is essential for several important cellular processes, but and EXCESS of copper can also lead to oxidative damage (what Shen.Ming Chen said as well)

"Why is copper more toxic to some cells than others?": different rate of UPTAKE of copper ? I would suggest to check  the uptake of copper for that cells are more sentive to copper and perhaps you should use less copper . Perhaps cells show different rate of copper uptake and the one are more sensitive to copper are taking up more copper and in turn leading a fast oxidative damage compare to other cells which are less sensitive instead.

The mode of action of Cu toxicity is actually far from fully understood and certainly includes binding to thiols and ROS productions but other processes can occur too (e.g. Fe displacement) that in turn might depend on the dose and duration of exposure.

I do think that - like Mario Grossi said - a different rate of uptake will be part of the answer between the different responses. However the detoxification mechanisms might be different for different celltypes (e.g. metallothionein production, excretion rate) and the number of processes that are sensitive to Cu might differ.