The difference really only matters for MyISAM, other storage engines do not care about the difference.
EDIT : Many users commented that InnoDB does care: link 1 by steampowered, link 2 by Kaan.
With MyISAM with fixed width rows, there are a few advantages:
No row fragmentation: It is possible with variable width rows to get single rows split into multiple sections across the data file. This can increase disk seeks and slow down operations. It is possible to defrag it with OPTIMIZE TABLE, but this isn't always practical.
Data file pointer size: In MyISAM, there is a concept of a data file pointer which is used when it needs to reference the data file. For example, this is used in indexes when they refer to where the row actually is present. With fixed width sizes, this pointer is based on the row offset in the file (ie. rows are 1, 2, 3 regardless of their size). With variable width, the pointer is based on the byte offset (ie. rows might be 1, 57, 163). The result is that with large tables, the pointer needs to be larger which then adds potentially a lot more overhead to the table.
Easier to fix in the case of corruption. Since every row is the same size, if your MyISAM table gets corrupted it is much easier to repair, so you will only lose data that is actually corrupted. With variable width, in theory it is possible that the variable width pointers get messed up, which can result in hosing data in a bad way.
Now the primary drawback of fixed width is that it wastes more space. For example, you need to use CHAR fields instead of VARCHAR fields, so you end up with extra space taken up.
Normally, you won't have much choice in the format, since it is dictated based on the schema. However, it might be worth if you only have a few varchar's or a single blob/text to try to optimize towards this. For example, consider switching the only varchar into a char, or split the blob into it's own table.
You can read even more about this at:
http://dev.mysql.com/doc/refman/5.0/en/static-format.html
http://dev.mysql.com/doc/refman/5.0/en/dynamic-format.html