The main benefit of adhering to the project schedule is that your project will be finished on time (as promised to the client/customer)!
Now, for the project team, if you usually adhere to the project schedule, then they will feel more relaxed working with you because they will know that their hard work will definitely bear fruit at a defined point in time (which is the end of the project). They will also be more productive.
For you, the project manager, adherence to the project schedule means more trust from your stakeholders and from your team members. It means that you're the man who "gets things done" and who sticks to his promises.
The project sponsor and the project stakeholders will be definitely satisfied when you adhere to the project schedule, simply because they don't want to keep meeting about the same project forever, and they want to get the project done so that they can benefit either directly or indirectly from the resulting product/service.
As for the company, consistently adhering to the project schedule will attract more clients (think about a contractor in the construction industry who always finishes work on time versus someone who always misses his deadlines - which one do you think the developers will send projects to), it will also enhance the ROI and maximize profits. Which in its turn means that everyone in the company will profit (raises, bonuses, etc...) including team members, project managers, and the stakeholders.