Both Russian sentences are conditional sentences with the protasis (the part describing the antecedent) omitted or implied.
It describes the consequence ("will not just sit and watch") of an antecedent which is not mentioned.
For the sake of discussion, I will assume that the antecedent is "a pandemic breaks out"
Левый госминистр не из тех, кто будет сидеть сложа руки (если случится пандемия)
This is an indicative (factual) conditional sentence.
It assumes that everything except the antecedent is in place. This means that we do have a Left Prime Minister, they have enough power to do something except sitting and watching, etc. Everything, except the plausible pandemic, is there, and we are making a logical conclusion from the known facts.
Левый госминистр не из тех, кто сидел бы сложа руки (если бы случилась пандемия)
This is a subjunctive (counterfactial) conditional sentence. It describes the consequence of the antecedent in an imaginary world, where it's plausible and the rest of the facts check out. In the real world, it is not so.
This sentence does not describe the real world. We might not have a Left Prime Minister, or the pandemic is implausible, or it's a reductio ad absurdum ("the Prime Minister says they're Left, but a Left Prime Minister would not just sit and watch, and this one is sitting and watching, hence they're not Left, q.e.d.")
It works almost the same way in English. English marks the subjunctive mood with the past tense of "will", Russian with a past tense with бы.
English is a little bit more forgiving about mixing the moods. The phrases like "if you feed a dog chocolate, it will die" and "if you fed a dog chocolate, it would die" can both be used in the factual meaning.
In Russian, если накормишь собаку шоколадом, она умрёт is factual, while если бы ты накормил собаку шоколадом, она бы умерла is counterfactual.
That said, we don't have enough context to translate this phrase. We don't know if the situation in question is factual or not, so we cannot pick the right mood.