As you probably know PHP "mysql" extension supported persistent connections but they were disabled in new "mysqli" extension, which is probably one of the reasons some people delay migration to this extension.
The reason behind using persistent connections is of course
reducing number of connects which are rather expensive, even
though they are much faster with MySQL than with most other
databases.
Not only connects are expensive but you also may run into the trouble establishing number of connections you need. The problem is there can be only so many connections active between Host "Apache" and Host "MySQL": Port 3306 as connection in TCP/IP protocol is identified by pair of IP addresses and pair of ports (local port and remote port). Yes if you're establishing thousands of connections per second you normally do not keep it open for long time, but Operation System does. According to TCP/IP protocol Ports can't be recycled …
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