Part of the laws intrinsic to the service include the uniforms, and regulations around them. The Kohen Gadol had extra clothing, with their own laws:
וְיִרְכְּסוּ אֶת הַחֹשֶׁן מִטַּבְּעֹתָיו אֶל טַבְּעֹת הָאֵפֹד בִּפְתִיל תְּכֵלֶת לִהְיוֹת עַל חֵשֶׁב הָאֵפוֹד וְלֹא יִזַּח הַחֹשֶׁן מֵעַל הָאֵפוֹד – They shall fasten the breastplate by its rings to the rings of the apron with a blue cord, so that it will be on the band of the apron; and the breastplate will not move off the apron. (28:28)
Although separate, the breastplate and rear-facing apron were fastened together at all times. Simply because the breastplate did not have a neck chain, and the apron had no shoulder straps – they would balance and offset each other. But the Torah is not giving logistical or fashion advice – if this is how they are worn, it need not be specified at all. Why emphasise that they are inseparable then?
The Gemara in Erchin explains how each of the garments the Kohen Gadol wore would atone for a different national deficiency. The apron atoned for idolatry, while the breastplate atoned for financial dishonesty, with regard to both business and judicial matters.
R’ Moshe Feinstein notes that this could very well be the reason that the breastplate and apron were inseparable – they share a common facet. Someone who worships idols does not believe that God controls all things. Someone who cheats, steals, distorts, or embezzles in their finances is guilty of the same crime!
Dishonesty, and all forms of financial impropriety demonstrate that the guilty party believes that both no-one is watching, and that they can get more than what ought to be coming their way. This is entirely heretical, antithetical to Judaism, and quite similar to idolatry.
R’ Moshe Feinstein explains that the root of both is the same – a belief that Hashem lacks control over the world. Therefore, since they are inherently similar, the Torah specifies that these two parts of clothing are inseparable- they are almost the same.