They can be two different weapons.
Hex Warrior (Xanathar's Guide to Everything, p.55)
If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon's type.
This means every weapon you create with the Pact of the Blade class feature benefits from the Hex Warrior ability. The "If you later" part of the rule is there to cover the fact you gain Hex Warrior at 1st level but gain Pact of the Blade later at 3rd level, if you choose it. It is intended to make the rule clearer, even though it probably achieves the opposite.
The rules do not preclude you from touching a different weapon, one that is not your Pact Weapon, at the end of a long rest and effecting it with the Hex Warrior ability in addition to every Pact Weapon you conjure. It is just that every Pact Weapon is effected, however else you use the ability.
Any magic weapon you make your Pact Weapon is included, Pact of the Blade (PHB p.108) states:
You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon.
as even if it is two handed the Hex Warrior ability "extends to every pact weapon you conjure with that feature, no matter the weapon's type".
As to Improved Pact Weapon (Xanathar's Guide to Everything, p.57):
the weapon gains a +1 bonus to its attack and damage rolls, unless it is a magic weapon that already has a bonus to those rolls
If you take "Improved Pact Weapon" the +1 bonus from this invocation applies to every pact weapon you create and, as it is a different effect to Hex Warrior, they both apply at the same time to a Pact Weapon.
However, as is explicitly stated in the invocation description, if you have made a magic weapon with a magical bonus your pact weapon as above, you do not get this +1, just the bonus from the weapon.
If the Hex Warrior weapon is not a Pact Weapon then the invocation does not extend to it.