In addition to Ryan’s answer, these words can have a more nuanced meaning.
Timeout often means stopping all activity, AND then the activity resumes. So a timeout is usually for a specific or short period of time. In your example, the timeout stopped entire game. All players stopped playing for a period of time, not just the injured player. Once the timeout was over, they started playing again. So it is a short period of time and not much happens during the time.
Moratorium often stops a very specific thing and is indefinite in length. In your example, ONLY testing nuclear weapons stopped. But people could still make nuclear weapons, or research better weapons, or sell weapons. They just couldn’t test them. And this moratorium would last indefinitely. Sometimes people say “temporary moratorium” to say that the moratorium will end soon. They have to specify “temporary” because a moratorium is not usually temporary.
An example of moratorium in everyday use: if a friend and I argue about something, like politics, we can decide that we will no longer discuss that subject. We could say, let’s put a moratorium on that subject, meaning we will no longer talk about it since it makes us argue.