First of all it is a good practice to use the list form of system or the qx// operator which does the same thing. Like this:

$status = qx(/usr/bin/tar cf $backupDir/$tar_filename tempDir);

$status = system qx(/usr/local/bin/gzip $backupDir/$tar_filename);
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My guess is the script is probably failing via cron because /usr/local/bin isn't on crons' default path. You should put a line like:

PATH=/usr/local/bin:/usr/bin:/bin
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at the top of your crontab and also for safetys' sake

SHELL=/bin/sh
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This works on Linux

Update: Hrmph, dws already answered this one. That'll teach me to write a post, go make a sandwich, and then click on submit. :-) But I should also mention there are Archive::Tar and Compress::Zlib(for gzip) modules on CPAN whereby you can avoid all this messy shelling to external programs.