C14 Nuclear Diamond to AA Battery Apples-to-Apples Engery output
Let me address this part of your question first:  


  "[...] how many Joules per gram would a C14 nuclear diamond produce if it weighed 20 grams"


I think it's important to understand that the bolded/italicized phrase is redundant, because your are discussing (as did the article) the energy output per gram.  So the per-gram output of a 20 gram battery is the same as that of a 1 gram battery.  The 20 gram battery would have a total output of 14000 Joules in it's lifetime, where the one gram battery would have a total output of 700 Joules.  The fact that the article, and your question, normalize to a per-gram basis is how they've maintained an apples-to-apples comparison.  

The semantics get tricky when they talk about the mass of the diamond battery, and by my thinking this is where they may fail to give an apples-to-apples comparison.  The diamond would not be pure $\ce{C^{14}}$, and at a minimum would contain a thin protective coating of non-radioactive $\ce{C^{12}}$ (protecting those on the outside from the radiation on the inside).  So when they say "A diamond beta-battery containing 1 gram of $\ce{C^{14}}$", it is not made clear what the total mass of diamond would be (e.g. we aren't given the ratio of $\ce{C^{14}}$ to the $\ce{C^{12}}$ that would also be present).  

That said, I think that the apples-to apples answer to your question, based on the information given in the article, is as follows:  

Conventional 20 gram battery: 14000 Joules/day, for 1 day.
Nuclear 20 gram $\ce{C^{14}}$ battery: 300 Joules/day, for 5,730 years.  

or, equivalently:   

Conventional 1 gram battery: 700 Joules/day, for 1 day.
Nuclear 1 gram $\ce{C^{14}}$ battery: 15 Joules/day, for 5,730 years.