Hello,
The scheduler documentation in the apex developers guide states:
You can only have 25 classes scheduled at one time. You can evaluate your current count by viewing the Scheduled Jobs page in Salesforce or programmatically using the Force.com Web services API to query the CronTrigger object.
while the batch apex states:
Up to five queued or active batch jobs are allowed for Apex.
So I’d say that they are independent, although the scheduled job starts batch apex or vice versa it will impact both.
Thanks.