Let the two numbers be a and b and assume that a is greater than or equal to b.

According to question, we can write the sum of the two numbers as

$a+b=1000$ ……….. (i)

And the difference between the squares of the two numbers can be written as

${{a}^{2}}-{{b}^{2}}$$=256000$

⇒ $(a+b)(a–b)=256000$

⇒ $1000(a-b)=256000$

⇒ $a–b=256000/1000$

⇒ $a-b=256$ ………….. (ii)

By solving equation (i) and (ii), we can find the two numbers

So, On adding the equations (i) and (ii), we get;

$(a+b)+(a-b)=1000+256$

⇒ $a+b+a–b=1256$

⇒ $2a=1256$

⇒ $a=1256/2$

⇒ $a=628$

Now, putting the value of a in equation (i), we get

$628+b=1000$

⇒ $b=1000–628$

⇒ $b=372$

Thus, the two required numbers are $628$ and $372$.