Because you are using Java 6, checking the modified date or file contents is your only option (there's another answer that discusses using the newer java.nio.file functionality, and if you have the option of moving to Java 7, you should really, really consider that).
To answer your original question:
You didn't specify the location of the file (i.e. is it on a local disk or a server somewhere else) - I'll respond assuming local disk, but if the file is on a different machine, network latencies and netbios/dfs/whatever-network-file-system-you-use will exacerbate the differences.
Checking modified date on a file involves reading the meta data from disk. Checking the contents of the file require reading the file contents from disk (if the file is small, this will be one read operation. If the file is larger, it could be multiple read operations).
Reading the content of the file will probably involve read/write lock checking. Generally speaking, checking the modified date on the file will not require read/write lock checking (depending on the file system, there may still be consistency locks occurring on the meta data disk page, but those are generally lighter weight than file locks).
If the file changes frequently (i.e. you actually expect it to change every minute), then checking the modified date is just overhead - you are going to read the file contents in most cases anyway. If the file doesn't change frequently, then there would definitely be an advantage to modified date checking if the file was large (and you had to read the entire file to get at your information).
If the file is small, and doesn't change frequently, then it's pretty much a wash. In most cases, the file contents and the file meta data are already going to be paged into RAM - so both operations are a relatively efficient check of contents in RAM.
I personally would do the modified date check just b/c it logically makes sense (and it protects you from the performance hit if the file size ever grows above one disk page) - but if the file changes frequently, then I'd just read the file contents. But really, either way is fine.
And that brings us to the unsolicited advice: my guess is that the performance on this operation isn't a big deal in the greater scheme of things. Even if it took 1000X longer than it does now, it probably still wouldn't impact your application's primary purpose/performance. So my real advice here is just write the code and move on - don't worry about it's performance unless this becomes a bottleneck for your application.