Probably.
Yes, differential is all changes since the last full backup.
But most third party backup software does backups that count as "full" backups. Some don't (doing file-level backups instead), but most of those also don't work very well with SQL Server databases.
Have you tried restoring from the third-party backups?
Have you checked the backup tables in msdb to see what the first and last lsn values are for the diff backups? That'll tell you more about what timespan it's covering.
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