There are four North American variants, but all came with five CD-ROM discs, a manual, and Silent Hill 3 Official Soundtrack CD. All six discs came in generic paper sleeves, but the sleeves are black or white depending on which variant.
There are two box variants, box 1 and box 2. The artwork is slightly different, but the boxes appear to be the same size. There are four disc variants (including the soundtrack), disc set 1, set 2a, set 2b, and set 3. Disc sets 1 and 2a came with box 1, while sets 2b and 3 came with box 2. Disc set 2a and set 2b may be identical, or just very similar. Either way, in total there are four possible combinations of boxes and discs.
The order of release is unknown, the given variant numbers are assigned only for practical reasons.
This page is for box 2, disc set 3, shortened to variant 2-3.
On this variant, the artwork on the front of the box extends above Heather's hair, showing additional piping. There is no mention of Vivendi Universal Games on the back of the box. On the bottom side of the box the barcode is on the right side. The box is, as far as I can see, identical to the box included with variant 2-2b.
Unlike all other variants, the discs do not have a ring of missing artwork 1 cm from the spindle hole. The artwork continues right until the center. Also note the placement of the copyright text on the soundtrack CD when comparing to other variants.
All the discs came in black paper sleeves, though the insides of the sleeves are white.
The manuals included with box 1 and box 2 are also different. The manual included with box 2 extends the artwork to a little above the hair. There may be a slight difference between the manuals included with variants 2-2b and 2-3, but I do not know yet.
The soundtrack included with the PS2 edition is yet another soundtrack variant. The easiest way to tell them apart is by looking at the placement of the copyright text on the discs.
1. Front
2. Back
3. Soundtrack CD
4. Variant comparison guide
Thanks to MobyGames (images 1-2) and Felix M. C. Li (image 3).