"Being" suggests it's temporary, whereas "were tiresome" can mean temporary or permanent. The nuance is a little more grumpy and negative. E.g. "Right now, the kids are great! But this morning, they were being tiresome." vs "I never want kids. My brother's kids, my god, they are so tiresome." "Tiresome" is a little old-fashioned/formal, however. You might want to say: * The children were being difficult. * The children were acting up. [A little casual, means to misbehave. This is a verb phrase, not an adjective.] * The kids were (being) a pain in the neck. [Very casual]

1- The children were tiresome - they were tiresome in general and not doing anything out of the ordinary from the usual, but taking care of children in general is tiresome. 2- The children were being tiresome - they were tiresome on purpose and agitating the person or being more tiresome than usual even if when you asked them to stop doing something, they continued to do things that made you tired and agitated. That's how I would take it. Number 2 is more negative.

Being is nothing but the present participle + ing. The first sentence "The children were tiresome" is where the tense is in simple past. The second sentence "The children were being tiresome" is where the tense is in past continuous which means that the children will eventually get tired. So in this case "being" is used to define the tense of the sentence.