Based only on the terms in the question, I can conclude that High Explosive shells are less effective at piercing armor, such as that found on battleships. Wikipedia confirms this.
An armor-piercing (AP) shell is a type of ammunition designed to penetrate armor. From the 1860s to 1950s, a major application of armor-piercing projectiles was to defeat the thick armor carried on many warships.
. . .
With the introduction of the first ironclads in the 1850s and 1860s, it became clear that shells had to be designed to effectively pierce the ship armour. A series of British tests in 1863 demonstrated that the way forward lay with high velocity lighter shells. The first pointed armour-piercing shell was introduced by Major Palliser in 1863. Approved in 1867, Palliser shot and shell was an improvement over the ordinary elongated shot of the time. Palliser shot was made of cast iron, the head being chilled in casting to harden it, using composite molds with a metal, water cooled portion for the head.
The remainder of that section is also instructive.
High Explosive (HE) shells are intended for antipersonnel use, and are defeated by armor. AP shells are required to damage a target protected by armor.
It appears that the Japanese did attempt to rearm.
Frantically the Type-3 incendiary shells were ordered switched to armor-piercing. A veritable "stampede" in the magazines resulted, for all knew that a hit on either battleship before the highly explosive bombardment shells were safely stowed could turn them into infernos. But by some providence the Japanese did not understand, Fortune gave them a reprieve of eight long minutes.
(Note that this source asserts that they were armed with incendiary shells, not HE). Either HE or HEI would need to be restowed properly because detonation of one of these shells inside the ship would have been catastrophic.
The same source continues,
This gave Abe the vital time he needed to largely finish the switch of his main battery ammunition (though some incendiary shells were reported by the U.S. Navy), and assess the situation. Though confusion persisted about the precise location of the YUDACHI, the Japanese, unlike the Americans, could be sure the targets in front of them were enemy. Six minutes after sighting, Abe gave the order to illuminate with searchlights and to open fire. The action began at 0148 (Local time---Tokyo time was two hours earlier) when HIEI in concert with destroyer AKATSUKI and others snapped on searchlights to illuminate Callaghan's approaching force.
HIEI unleashed a truly devastating opening salvo, apparently hitting ATLANTA at the outset. Caught in the illumination, the ATLANTA and the SAN FRANCISCO behind were all but sitting ducks, spotlighted for all the Japanese fleet to see. But they were far from impotent, and were almost at once returning fire with equal gusto. Thus began what historian Paul Dull and others have referred to as "the most confused, close-ranged, and horrendous surface engagement of the war." It was also surreally brief. In reading of the detailed hard-fought action in multi-paged accounts, it is both startling and easy to overlook noting the fact that the time spanned from when HIEI and AKATSUKI snapped on their searchlights to the end of the action was barely a half hour.
At least this source seems to believe that the type of shell was not the critical factor.