Pros and cons of using an external MySQL database?

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Hi All,
I am working on a site where I may have to use an external MySQL database, in other words a database that is not with the same host.
I have no experience with this and have read this can be done quiet easily by using the database IP address.
I would however have a few questions:
-Security: is it as secure as using an internal DB, I would be working from an HTTPS domain thus the info exchanged with the server would not necessarily be of a very sensitive nature.
-Speed: if there a noticeable difference in speed, again as opposed to an internal DB
-Errors: i read somewhere that there can be difficulty from time to time for the connection to be established, any experience?
-Can I restrict the IP address from the referrer accepted by the database
Any other pros / cons any one may have would be of assistance.
Thank you in advance.
Some issues you will face, although it is done frequently:

1. There will be a latency issue but depending on the data it may not be noticeable
2. MySQL connections tend to be clear text by default - so you need to take care. Just sending people's e-mail addresses for example, are stricly governed
3. Errors/Speed - try to keep it stateless (i.e. don't expect you have connection, be prepared to open it with each query)
4. HTTPS is between the web host and the client user (Browser <-> web server). You won't be connecting on HTTP/HTTPS protocols generally speaking, to the MySQL Server

Settings like whether or not you can use SSL data transfer, which IP's may access it are down to the admin and not generally open to modification on shared hosting platforms.
I think link8r covered it but I just want to re-iterate, that you should be wary about the type of data your transmitting over open 'space'. If its personal data like names, DOB, emails, address, or even a payment process you will need to have sufficient security in place like SSL to protect this data as you don't want to be open to legal action should people's personal data get into the wrong hands.