To write a Python script that efficiently outputs all prime numbers inclusively between two numbers, you can implement the Sieve of Eratosthenes algorithm. This algorithm is known for its efficiency in finding prime numbers within a given range.

Here's the script:

```python
def sieve_of_eratosthenes(limit):
    primes = [True] * (limit + 1)
    primes[0] = primes[1] = False

    for p in range(2, int(limit**0.5) + 1):
        if primes[p]:
            for i in range(p * p, limit + 1, p):
                primes[i] = False

    return [p for p, is_prime in enumerate(primes) if is_prime]

def primes_between_numbers(start, end):
    primes = sieve_of_eratosthenes(end)
    primes = [p for p in primes if p >= start]

    return primes

if __name__ == "__main__":
    num1 = int(input("Enter the first number: "))
    num2 = int(input("Enter the second number: "))

    primes = primes_between_numbers(num1, num2)

    print(f"Prime numbers between {num1} and {num2} (inclusive):")
    print(primes)
```

This script first implements the Sieve of Eratosthenes in the **`sieve_of_eratosthenes`** function to find prime numbers up to the given limit. The **`primes_between_numbers`** function then filters the primes to include only those within the specified range. Finally, the script prompts the user for input, and outputs the prime numbers between the two given numbers.
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