One reason a Western blot is more specific than an ELISA - even one using the same set of antibodies - is background.
Antibody-linked colorimetric reactions aren't completely on/off. There's always some background reaction. The antibodies are a little bit cross reactive, or non-specifically sticky, or the colorimetric coupling enzymes aren't completely washed away, etc. There's a number of reasons you'll get a small amount of signal even in the abscence of the desired reagent.
So imagine how this non-specificity looks in an ELISA assay. You're only getting a single readout. Any non-specificity/background in the assay adds noise to your output level. Hopefully for any decent assay the noise is normally well below the threshold for crossing over into a "positive" readout, but if the noise is Normally distributed, there's always a small chance that the noise will exceed that threshold. (There's also those non-normal failure modes, where you mess up a wash step, or you get a bad tube of antibody, etc. Negative controls help, but only so much.)
In contrast, think about how non-specificity/background looks in a Western blot. If you get non-protein linked background, the presence of this background will be across the entirety of the blot. It's very obvious that something went wrong with the assay in those cases. Even if you get random spots and not overall background, the chance you'll get a random spot at just the correct molecular weight is very, very unlikely.
An additional benefit is that Western blots first separate the proteins prior to doing antibody-linked detection. This means if there's some protein in the sample which cross-reacts with the antibody, it's very likely it will migrate at a different molecular weight and not be counted as a false positive. (As opposed to ELISAs, where every signal counts toward the total.)
These effects mean that if you're looking for low false positives (as you would with an HIV test) a well-done Western blot is likely to give a more confident signal than a well-done ELISA would. (At the cost of time and effort.)
A final issue is that combining the two will give you more confidence in a positive result than either one alone, especially if you use different antibodies for each. The final confidence in a positive result is not just the confidence from the Western blot, but the confidence that both the Western and the ELISA report positive results, so there's a benefit of doing both, even if they were equally specific.