TL/DR: In colloquial English, you should be fine using simple past tense to describe the reason that something occurred, as it can be considered concurrent (and thus doesn't violate the requirement that events must follow in order to use simple past).
Let's take your questions in the order they appear:
Do the two examples differ in meaning?
No, they do not. And colloquially, both sentences are correct as many English speakers will use drank for both simple past and past perfect.
Do we...need to use a past perfect in the subordinate clause of a sentence like this?
In colloquial English, no. The argument could be made for both simple past and past perfect.
One one hand, the effect is being described before the cause (feeling ill is coming before the drinking of the coffee), thus breaking the rule of simple past's first-to-last order. This would mean using past perfect.
On the other hand, the subordinate clause is describing a reason for the independent clause (drinking that much coffee MADE me ill), thus can also be considered concurrent. This, of course, would mean using simple past.