Most MySQL users know and have some experience with the web-based phpMyAdmin 'frontend' for MySQL. We even sometimes (not quite infrequently actually) experience that users think that phpMyAdmin and even the PHP language is an integrated part of MySQL itself. But that is not the case at all.
There is nothing bad about phpMyAdmin. It is an almost complete data manager for MySQL. And being a web-based tool you can have it available from any server running the LAMP or WAMP software 'stack'. You do not need to have anything installed on your local machine. That is the advantage of it.
With SQLyog you won't have to wait for annoying page-refreshes of your browser, and you will see your productivity increase rapidly. Also context-sensitive menus and inline-help is immediately available. And a full-featured set of user-friendly keyboard-shortcuts as well. And much more of the stuff than it will ever be possible to implement in a web browser.
Also from time to time we hear from users something like: 'My ISP tells me that they do not allow connections from outside. I will need to use phpMyAdmin'. That is sheer ignorance on behalf of that ISP-person saying so! SQLyog changes the rules of the game. Just by uploading a single PHP page to your web server, you can use all the cool features of SQLyog. The PHP page exposes the MySQL API as a set of web-services that can be consumed by SQLyog. Even when connected to a remote mySQL server with HTTP you will experience much faster operation than any browser-based tools can do. And SQLyog allows for safely encrypted HTTPS- and SSH-based connections too.
In addition SQLyog Professional, Enterprise and Ultimate gives you 'powertools' that you won't find similar in phpMyAdmin and an 'auto-complete' feature, that is quite unique.