let's say that the category with the products has the id 10.
And you want to copy all the products to category with id 20.
Run this script.
$sourceId = 10;
$destinationId = 20;
$source = Mage::getModel('catalog/category')->load($sourceId);
$destination = Mage::getModel('catalog/category')->load($destinationId);
$products = $source->getProductsPosition();
$destination->setPostedProducts($products);
$destination->save();
But keep in mind that any product you have in the destination category will be removed from that category.
If you want to keep the products you already have in the destination category use the following.
$sourceId = 10;
$destinationId = 20;
$source = Mage::getModel('catalog/category')->load($sourceId);
$destination = Mage::getModel('catalog/category')->load($destinationId);
$products = $source->getProductsPosition();
$destinationProducts = $destination->getProductsPosition();
$destination->setPostedProducts(array_merge($products, $destinationProducts));
$destination->save();