Following comparison of world's two most popular database management systems (DBMS) is based on their use as a database for making web based applications.

Now that MySQL is officially a part of Sun, i would any day prefer MySQL for making my web based applications than any other huge (read heavyweight) databases, which probably offers so many features that are basically not needed for creating such applications.
Read Jonathan Schwartz's Blog to find more.
As regards Comparison, i would like to highlight following points:
Firstly being Open Source, MySQL comes as a free download for developers worldwide. On the other hand, for using Oracle, you need a license (and for that matter a pretty costly license)
With Google, Facebook and Sina.com prefering MySQL for creating network services, one cannot doubt over MySQL capability of handling high volumes of data (perhaps in terabytes or even more)
MySQL is already known for its faster data retrieval, and impressive clustering and replication features and with MySQL 5.0 supporting stored procedures, views and triggers, the situation has only improved.
If you are a newbie to this world of Databases, i would recommend you to start with MySQL and then gradually shift to some other, if need be.
Its not that Oracle is not useful or because i'm a SUN CA, I will continue to praise Sun's technology and products. 

The truth remains that Oracle provides a huge array of features as compared to any other existing DBMS, let alone MySQL. Oracle is considered as a giant in the world of database installations.
For these reasons, it is heavyweight and installation and uninstallation can get a bit tricky for novices.
In a nutshell, features is surely one area where MySQL loses to Oracle, but then MySQL is devoid of various disadvantages of Oracle.
Also Oracle 9i and 10g both can also be used a ORDBMS(Object Relational Database Management Systems) and supports ORSQL which only a few other databases like Informix and IBM's DB2 does, but MySQL cannot be used to implement a ORDBMS design.
To summarize, i would say that In the world of databases, there is no "one size fits all".
If you need the Oracle features, MySQL isn't going to work for you.
Conversely, if you don't need the Oracle features(which is most often the case), MySQL is a lot cheaper,
even with the license and the support contract.
And with Sun acquiring it, the situation is only going to improve. 
Top Exec Speaks Up
Mr. Charles Phillips, president of Oracle, once made a statement at some conference about the comparison between Oracle and MySQL: "We're both in the transportation business," Mr Phillips said. "We have a 747, and they have a Toyota."
Mr. Marten Mickos, CEO of MySQL is more than happy with the comparison made, since there are many more Toyotas sold than 747s. "Toyota is a very profitable company," he added. 
Mr Mickos maintained that the company is still more interested in new web-based application customers than it is in replacing existing database installations.
(taken from http://www.cbronline.com/article_news.asp?guid=9231B8BD-3788-4DB2-B85F-707E75857B58 )
Disclaimer:
The above comparison is based on my personal experience with both DBMS. If you find some missing/wrong information, please leave a comment.