Stability of diboron(B2) and dilithium(Li2)
Firstly, the use of molecular orbital diagrams allows you to predict the electronic configuration, if a molecule would exist or not (as you say, evaluating the bond order), and some magnetic properties. Moreover, if you have "the same molecule" as O$_2$ and its ion  O$_2^-$, etc, you can say the higher bond order, the bigger stability. 

But if the bond order is the same (your case), you can not talk about stability using those diagrams. 

Coming back to the question, looking the dissociation energy (the energy relative to separated atoms) for B$_2$, you will see it is bigger in module. 
So it will be more stable. 
I do not have a qualitative way to estimate which of those molecules is more stable.

The way to know it theoretically is using electronic structure programmes.