Obviously whoever built this was not very good at what they were doing and/or not really paying much attention to detail. When working with security critical things like authentication, that's not great.
Think of it like this: If you found that your electricity guy had accidentally installed the outlet sockets upside down, that in itself isn't the end of the world. But wouldn't it make you a little bit worried that he messed up something else more critical and your house will burn down?
Also, short limits to password length might be an indicator of a legasy system with no or bad hashing. Even worse, as jrtapsell highlights, the fact that the password is accepted when set but not at logg in could be a sign that it is truncated when inserted into the database and hence stored in plaintext.
So while this is proof of nothing, it should still set off your alarm bells.
thisismyspecialpasswordduring registration, but it only allows 4 characters during login, enterthis. – Dec 07 '17 at 14:07