What advantages do either method offer for html, css and javascript files served by a LAMP server. Are there better alternatives?
The server provides information to a map application using Json, so a high volume of small files.
See also Is there any performance hit involved in choosing gzip over deflate for http compression?
The main reason is that deflate is faster to encode than gzip and on a busy server that might make a difference. With static pages it's a different question, since they can easily be pre-compressed once.
On Ubuntu with Apache2 and the deflate module already installed (which it is by default), you can enable deflate gzip compression in two easy steps:
a2enmod deflate
/etc/init.d/apache2 force-reload
And you're away! I found pages I served over my adsl connection loaded much faster.
Edit: As per @GertvandenBerg's comment, this enables gzip compression, not deflate.