Sander: Thanks for taking the time to review and respond to my question, but I no longer need help at this time since encapsulation with $N()$ provides a satisfactory workaround for the problem. If you're interested in the general topic of prime counting functions, you might be interested in some of the results which I've illustrated on the following website.
Fourier Series Representations of Prime Counting Functions

I've used this form of a Show statement which combines a Plot with a ListPlot successfully with the invocation of other simpler functions, and I suspect the complexity of the particular functions being invoked here is exposing a bug in the Mathematica implementation. The function being invoked shouldn't matter. I shouldn't get a different result when using ListPlot than I do when using DiscretePlot. I modified the Show statement with an invocation of N[] around the dOfZetaAlt2Opt function as follows and now I get the correct results, so perhaps Mathematica is getting confused while trying to keep track of too many exact results.

Show[Plot[{Im[dOfZetaAltOpt[1/2 + I x, 1/2, 10, 10]], 
   Im[Zeta'[1/2 + I x]]}, {x, 1, 25}, PlotRange -> Automatic, 
  GridLines -> Automatic, PlotPoints -> 200, MaxRecursion -> 0], 
 ListPlot[Table[{x, 
    N[Im[dOfZetaAlt2Opt[1/2 + I x, 1/2, 10, 10]]]}, {x, 1, 25}], 
  PlotStyle -> {Red}]]