This change seems like a licensing issue, but is still undesirable and needlessly changes the episode featuring the 'What a Wonderful World' song. I can't believe the song was completely unavailable for licensing. So your guess is as good as mine as to why Pioneer omitted it from the English audio.

At least one overlay has been identified in Vandread.

In episode 3, a stuffed bear is sitting in a chair on the bridge,and there's a sign on the bear saying "reserved." You can tell this is an overlay without even having the R2 DVDs since the exact same shot is in a Japanese promo in the extras, with Japanese text. The original Japanese says "yoyaku-seki"("Reserved seat" would be a better translation and make more sense in this situation).

Asgirl2040 provided this information on a change in the opening used during Vandread:Second Stage:

A small EDIT occurs on volume 2 of Vandread: The Second Stage. It only affects one episode so far, but will, by its nature, affect all remaining episodes as well.

First, an explanation:
Aside from the ever-changing episode-specific shots, there are two versions of the OP sequence for Vandread: The Second Stage. The first version is used for episodes 1-6, and the second is used for episodes 7-13. Fuzzy shots become clearer and/or more detailed, the sequence of character shots that comes up just before the show logo is different, the sequence of character shots that comes up about halfway through is replaced by a sequence of Pyoro, Misty, Kahlua, and Ezra, and so on. Watch the credit-less OP sequence for episode 6, and then the one for episode 7, as an easy way of seeing the differences.

Second, the problem:
One of the changes for the second version is the split screen sequence of all the Vandread transformations (during which the chorus of the OP is playing). The shots are clearer and are actually the slightly tweaked versions used throughout VSS, while the first version consists of the original transformations and is rather fuzzy and indistinct. The credits that appear during this sequence are the credits for the OP and ED songs. While Pioneer left most of the original credits in place, due to the accompanying English credits used by Gonzo, they did have to replace a couple of credit screens that were entirely in Japanese with new English credit screens. One of these was the OP and ED song credits screens.

Third, the edit:
Pioneer took this credit screen from the first version of the OP sequence and used it in the second version, in lieu of typing up a new credit screen. What this means specifically is that the following sequence of shots from the second version of the OP sequence was REPLACED with the same sequence from the first version:
Shot of the Nirvana, shot of Hibiki's Vanguard, split screen shots of Vandread transformations.
An easy check is to watch the credit-less version of the OP sequence for episode 7, then watch the actual OP sequence in the episode.

Fourth, the reason:
One can assume that Pioneer felt REPLACING this sequence was easier than typing up a new credits screen to overlay on top of the second version.

The original Japanese track & subs are uncut on the DVD. But the BGM (Meia's music box plays an instrumental version of "What a Wonderful World" and the piece is used quite a few times during the episode with & without the box) and ending song (a female singer who sings 'What A Wonderful World' in English) have both been replaced with a new song in the English audio.

When Meia is given the music box as a present from her Ohma (her female father figure), in the original Japanese (and in the subs) she is told the song playing is called "What a Wonderful World". In the dub, the new song is called "Some Days". The ending song in the dub is the full vocal version of this new song.

It also appears that the original TV broadcast used the song as sung by Louie Armstrong, while the R2 DVD used the female singer that also was used on the R1 Japanese audio.