Always remember that Book of Exalted Deeds is a poorly-considered, poorly-written, poorly-edited book. Even obvious gaffes like this are none too surprising, considering the many, many problems it has.
There are three interpretations of this line, two of which hinge on assuming that a mistake was made—and the third of which is so broken that it’s an even larger mistake, albeit in judgment rather than mechanics.
They forgot apostles of peace require Vow of Poverty, or added that requirement later and forgot about this line. Simple, glaring editing mistake.
They were thinking of the rules in Dungeon Master’s Guide that say you need only meet prestige class requirements in order to take the 1st level, and imagining an apostle who broke their Vow of Poverty and saying even then, they still have some restrictions. Except that means they forgot that Book of Exalted Deeds specified that the prestige classes in that book lose their class features if they
lose their requirements actually, they only lose their class features if they turn evil, so this actually might work out. Very unusual situation, not at all clear if this is what they meant. Still calling this really poor writing.
They intended it to be an exception to the Vow of Poverty rules. This avoids any original contradictions in the rules, but one the writing is unclear, and two, more importantly, that is pants on head crazy. The apostle of peace is so overpowered that it actually justifies Vow of Poverty, one of the most crippling feats in the game. Letting them then get out of most of the limitations of that feat is just insanity.
So you can pick which interpretation you like best—the second one probably makes the most sense, since it basically says you can be an apostle of peace who breaks the Vow of Poverty, but you still can’t wear armor, and that’s... better, sort of, than the other options. But it should all be moot because you should never allow the apostle of peace, or any of its required feats, into your game. All of them will do terrible things to your character and/or game.