What 3rd party database backup tool are you using and where is it writing the backups to?
I personally like having my backups available for me to restore to test systems and so I can verify that the backup was successful. A backup is only known good after a restore, so I'd be sure that you can restore as you expect from the new tool.
But to answer your question, things I'd be concerned about are:
1 - restore times with the new tool vs native backups?
2 - are point in time restores possible?
3 - can you restore a database with a new name?
4 - backup times with the new tool vs native backup?
5 - how does it handle system database restores?
Basically, anything that can impact your RTO and RPO I would want to know. Would also be good to know the retention plan for the backups. Depending on what the tool is doing with backups, you may want to do backups to disk as well. For example, if the backup tool is just grabbing the mdf and ldf files and putting them on tape backup, I'd want a native backup as well. If you need to restore to a specific point in time, I don't want to restore to risk restoring to a point where data in memory doesn't match data on disk. I want to know that a checkpoint was performed prior to the backup starting. Plus if it is grabbing the MDF and LDF to put them on tape (I am just making an assumption about the backup tool), it could very easily lock the file while it copies it over which is a HIGHLY undesirable place to be...
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!