The issue is not whether or not you allow the simulacrum to attune to items independently, rather, it is which items you allow the simulacrum to make use of.
Generally speaking, the difference will be small. The simulacrum's increase in power will be trivially equal to the increase in power those items would provide to the original caster, since the simulacrum and the caster share game statistics.
However, the spell puts some restrictions on the simulacrum's abilities, and certain magic items may or may not interact with those restrictions, depending on the DM's ruling. So it is not so much if you allow the simulacrum to attune to items independently, rather, it is which items you allow the simulacrum to make use of. Here are the relevant restrictions from the spell description:
The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.
I won't provide an exhaustive list of magic items that might interfere with these restrictions, but I will give a few motivating examples. Consider the Pearl of Power:
While this pearl is on your person, you can use an action to speak its command word and regain one expended spell slot. If the expended slot was of 4th level or higher, the new slot is 3rd level.
While it is to me clear enough that the Pearl of Power would not work to restore spent spell slots to a simulacrum, there is room for a DM to rule otherwise (through a somewhat ill conceived reference to specific-beats-general). Should the DM rule that the Pearl of Power works, this is a significant buff to the simulacrum.
The other consideration is items that provide magical healing. This is discussed in detail in this Q&A: Can the creature created by a Simulacrum spell be affected by healing magic?. In my opinion, Rubiksmoose makes the most compelling case when he writes:
Allowing other forms of healing means that the alchemical process would be useless
This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way for a simulacrum to regain hit points.
It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features that let you heal.
To me, it is abundantly obvious that the alchemical process outline in the spell description is the only way to heal a simulacrum. However, wax eagle outlines an argument to the contrary. For example, consider the Ring of Regeneration:
While wearing this ring, you regain 1d6 hit points every 10 minutes, provided that you have at least 1 hit point.
If the DM allows this ring to work on the Simulacrum, then the expensive healing process typically required is entirely bypassed.
So you should discuss with your DM before hand about how to rule on items that interact with the restrictions contained in the spell description. If you allow those restrictions to be bypassed, then you make an already strong spell significantly stronger.