The difference here is that the Shroud of Turin is a remarkable subject manner even in a secular sense.
It is very important to consider the more recent history of the shroud when trying to determine its authenticity. For centuries, the shroud was very similar to a lot of those other "relics" that you described -- it generated a lot of interest among religious people, but not much more than that.
This all changed late in the 19th century when an Italian photographer named Pia Secondo was given permission to photograph it during one of its infrequent public displays. When he was developing the film, he made a startling discovery: the faint reddish-brown image on the light-colored shroud contained an incredible amount of detail when seen on the photographic negative. Secondo recognized the implication of this discovery immediately -- The image he was looking at on the negative bore all the markings of a photograph, and the shroud was actually the photographic negative.
The Shroud of Turin immediately began to attract a lot of attention in the scientific community, because nobody truly believed that someone could have created something like this 500 years before photography was even invented. And yet that is precisely what would have had to happen if someone had fabricated the shroud as a hoax.