Dismissal with notice is a legal dismissal and will start as per the date mentioned in the notice. Such kind of notice is given generally when employment contract is about to get over in the near future. The purpose of this notice is to inform the employee about end of contract.
Many times contract of employment involve the clause of the payment in lieu of the notice period. In such case employee can take payment for the notice period and end the contract. This will be the case when employer wants to end the contract. There may be similar clause from the employee side also which allow payment in lieu of notice period.
Dismissal with notice is legal termination of contract and dismissal without the notices will be acceptable only in case of the gross misconduct with proper investigation, hearing etc.