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Why prices don't update
So you've noticed that your prices differ from the target website's.
1. In most cases, that's normal. In some stores, prices can frequently change, but External Importer syncs products on a fixed schedule.
Go to the product edit page to check the time of the last update. You can also force update the product by clicking
Sync now. 
2. Check for HTTP and other errors in the product request. Your server may have been blocked, or the parser couldn't extract data for some other reason.
Check if the domain isn't currently throttled.
If this is the case, you can find the exact reason for it in
External Importer > Settings > Extractor. This can be: exceeding the daily request limit, or too many consecutive errors were received.
3. Check warnings and errors by going to
External Importer > Logs.
4. Check your synchronization period settings by going to
External Importer > Settings > Synchronization. Also, make sure that price synchronization is enabled.
5. If you use Cron synchronization, try switching to Frontend synchronization and refresh the page. Did the price update? Cron may be incorrectly configured for your WordPress installation. Try disabling and enabling External Importer to restart all of the plugin's cron tasks.
You can also install any cron manager plugin, like WP Crontrol, to check what's happening with cron tasks.
Please note that your website must have traffic to run cron tasks regularly. If there's no traffic on your website, you can replace WordPress cron with a real Cron job.
6. Cron synchronization can update a limited number of products. The number of products on your website may be out of proportion to the update period.