Is there any difference on backup process whether it is recovery model is Full or Simple? In past i have done server migration and i moved my databases by backup via this script template.

BACKUP DATABASE [ExampleDB] TO  DISK = N'D:\Backups\ExampleDb.bak


And i restored bak files on new server. Unexpectedly a user reported data lost after migration and i validated it by checking data in old server. It was around 8 hours data lost. This was very unexpected moment for me. 

My question is having backup via that script will give me full backup whether it is in Full or Simple recovery model ? 

In another way, Is there any difference between backup files when i get backup whether recover model is Simple or Full ?
A full database backup will give you everything in the database regardless of recovery model.  Upon restore, any transactions that were still in-flight when the backup completed will be undone.

Recovery model really comes into play during restores.  

With SIMPLE recovery, you can only take full and differential backups and that is the granularity that you can restore.

With FULL recover, you still take full (and optional differential) backups, but you also take transaction log backups and can use the combination of the full, diffs and logs to restore to a point in time.