Dynamic linking provides some benefits: You save disk space and some memory, especially for things that are used by most of the system like libc. You gain the ability to fix some problems in libraries without updating every app that uses the library.

Dynamic linking as implemented by most UNIX-Like OSes has a huge drawback: you only have one system copy of any given library, so all your applications are stuck using the same copy. There's no easy way to have multiple versions (say, with different features) of the same library without giving them different names. This isn't really a failing of dynamic linking, it's a failing of the OS library management model.

Dynamic linking in general has some other drawbacks: you lose Link-Time Optimization, there's more overhead at load time (and on the first call with lazy linking), and a few other performance-related issues can crop up on occasion.

There's also the issue that replacing a dynamically linked library is a lot easier than replacing a statically linked one for an attacker, so dynamic libraries can allow for vulnerabilities to be introduced (though this also is an OS issue, and there are often good mitigations for it).