Let's say I have a Sunday full backup and Monday through Friday differential backups, and I restore to Monday's differential backup by restoring the Sunday full backup followed by a Monday differential. If some of the transaction log backups between Monday and Friday are corrupted or lost, something goes wrong on Saturday, and I need to restore to the Friday state, do I still need to do a full restore plus a Friday differential?

Differential backups are incremental since the last full backup; Monday has all changes since Sunday, as does Tuesday, and so on. So to restore the database to Friday's state, you would restore the full backup, the latest differential (in this case, the Friday differential), then any transaction log backups since the differential. Differential backups are a great way to protect against problems with the transaction log backup, including damaged media, and to avoid the pain of having to restore hundreds of log backups. Microsoft recommends that before you restore over existing data, always back up the tail of the log (the records in the log that have yet to be backed up), just in case of problems with the restore.