Learn French: Le chat dont je parle ronronne, tandis que ton chien aboie ! - The cat I am talking about purrs, while your dog barks!

In French, pendant que is only used for time (meaning two things are happening at the exact same moment). Tandis que (and alors que) are used to show contrast or opposition between two ideas, which is what is happening here: purring versus barking. Translating it as "whereas" can help you remember the difference!

Could I use alors que instead of tandis que in this sentence?

Yes! Tandis que and alors que are very similar and both express contrast ("whereas" or "while"). Tandis que can sound slightly more formal or literary, but they are generally interchangeable in everyday French. You could easily say, "Le chat dont je parle ronronne, alors que ton chien aboie !"

Why is it dont je parle and not que je parle?

The verb "to talk about" is parler de. Because of that de, you have to use the relative pronoun dont (which replaces de + an object) instead of que (which replaces a direct object). Literally, it is like saying "The cat of which I am speaking."

I notice aboie ends in "-ie". Why is it not spelled with a "y" like the infinitive aboyer?