I am considering one of the following IT Management certifications: Scrum or ITIL. Could someone explain the difference between them, and which one is more relevant for IT Managers?
Redirecting the Scope of Your Question
[C]ould someone explain what is the main difference between those two certifications, and which one is more significant for IT Managers?
Specific career advice is off-topic as it's too subjective. In addition, this site is specifically about the field of project management, not IT management in general, and therefore the focus of your question is outside of the scope of this site. Furthermore, the value of any certification is non-deterministic; this related answer provides some guidance on how to evaluate certification value as it applies to your own specific circumstances.
Despite the foregoing, understanding the difference in purpose and intent of the two frameworks is useful. I differentiate the frameworks below, and provide references and links so you can research the differences between the frameworks further.
The ITIL and Scrum Frameworks
Wikipedia defines the frameworks as follows:
ITIL, formerly known as the Information Technology Infrastructure Library, is a set of practices for IT service management (ITSM) that focuses on aligning IT services with the needs of business.
Scrum is an iterative and incremental agile...development framework for managing product development.
The former is a framework for service delivery, while the latter is a product development framework. Service desks and operations management might concern themselves with ITIL, but it is unlikely to have central relevance to a project manager because projects are temporary endeavors rather than ongoing services.
If you are working as a project manager, then in my opinion, Scrum would be more appropriate than ITIL, as it relates directly to project management for a specific style of agile projects, but please don't forget that Scrum isn't the only way to run a project. You may also wish to consider Prince 2 or PMBOK to give you a wider perspective on project management, although as the tags you have selected for this question imply that you are working in an agile environment, they may be less relevant. Having said that, I believe they can be combined with Scrum, although I have not personally had experience of this, or seen it work effectively.
ITIL is less about project management, in my view, and is more about the wider aspects of managing IT. Admittedly, there are aspects of ITIL that will determine how you implement projects - change management, configuration management, and so on - but first and foremost, if you are a PM, then get a project management qualification. Then if you need some wider context, get yourself trained in ITIL later.
Set Certification Goals each Year
Is your organization more focused towards work involving ITIL? E.g., does your company have contracts that have ITIL v3 certification requirements? If so, having an ITIL certification first may give you an advantage in this area. It is a good practice to have knowledge of these requirements and set goals each year to improve your credentials. For instance, I recently just received my CSP (Certified Scrum Professional from ScrumAlliance.org) and will be taking the PMP (Project Management Professional from PMI) later this year. Instead of just getting one, why not get both?
On the CSM
The CSM certification now has an exam component, however this was implemented only recently and most folks in the industry do not typically hold it in high regard. With that said, consider the PMI-ACP as another option and one that carries a bit more weight than the CSM.