This question bothers a lot of our customers. There are three major effects switching from a local-only account to a Microsoft account:
Microsoft will be able to identify the users from within different applications, websites and services. Even if they are using other hardware after a successful login (e.g. mobile phones or your friends computer).
This identification might lead to a advanced profiling which may cause highly personalized ads. The same effect can be observed on other OS ecosystems with advanced cloud technologies like Apple iOS and Google Android.
Microsoft is using Windows 10, Office 2016 and other products of the current line-up to push their cloud services. In case of Windows 10 this includes synchronization of settings between devices (e.g. theme, browser settings).
This might also include passwords (browser, WiFi) which would increase the risk of breaches. However, Microsoft is using transport encryption and I assume they are also using some kind of encryption for the stored data. This would limit the risk of an incident. But still: Your sensitive data is leaving your local storage which increases the attack surface.
The same goes for OneDrive which is used to sync files between devices (like Dropbox). You are able to define OneDrive as your main storage which causes additional security-related and in some cases even legal-related implications. Under some circumstances and in some sectors this might not be allowed. This is one of the problems Swiss financial institutes are facing with Windows 10.
Some features and services require a Microsoft login to be used. This might include the Windows Apps Store and Cortana. Using these services might introduce additional risks of profiling (Personal Ads), data leakage (Cortana phones home) and further exploit vectors. But this is nothing new and also part of every other OS and online service. I don't have good example (yet?) where an exploitation of Windows 10 was only possible because of a Microsoft login.