What are the performance implications of having a web server (apache/php) and a database server (mysql) in completely different locations with different hosts.
Yes, there are performance implications, and they are all negative. Such a system could work, especially if the two systems are connected by a VPN with a Service Level Agreement based on both uptime and performance. If you are just going through "the cloud", you will have no consistency of network performance, and no one-throat-to-choke if connectivity or performance goes down.
just wondering if the network speed is significant when compared to the typical speed of database interactions.
That would be totally dependent on your network speed and your database and application.
There are two network performance aspects to consider- latency and throughput. You need to consider both.
For example, users will often be patient (up to a point) in waiting for a search query. They will not be patient for each page to load as they navigate a site. While both are viewed as latency from the user perspective, the first example might be weighted by the network throughput between database and web server, while the second by the network latency.
Or not! Only testing will tell.