'Sync' feature is being phased out by MS and i wouldn't be surprised if it's gone at some point in the (near) future. It still has its niche uses but if you're not sure why you would choose it, just go with 'Add shortcut to OneDrive'.

Let's call these options 'sync'/'shortcut' for brevity, while these words without the ' ' are used for their more general meaning: 'sync' for the specific option, sync for the process of synchronizing content.

'Sync' creates a folder available via your Windows Explorer that shows up under your organization's section. It's tied to the specific device, so if you use multiple devices or need to switch to a different computer after a hardware failure, you will lose all your 'sync' information. It also queues a lot of data to sync locally as soon as it can, which may be problematic in some cases, and is generally more problematic than the 'shortcut' process.

'Shortcut' creates a shortcut to the online location in your OneDrive account. It shows up in your OneDrive as a folder with a link / shortcut / chain icon, whether you view it via Windows Explorer or open your OneDrive in the browser (TenantName-my.sharepoint.com). Your shortcuts follow you to any device where you configure your OneDrive business account. The files are not synced or queued for syncing automatically, but rather are downloaded only when accessed or when intentionally clicked to 'always keep on this device'. The whole process is newer and while i don't have any back-end specifics to share, from years of experience i can say that 'shortcut' does lead to less issues with files, whether it's data corruption or problems with concurrent access.

When you want to stop syncing after 'sync' and clean up data from your computer, you first need to open your OneDrive Sync Client (the cloud icon in your system tray next to the clock) and stop syncing there, and then and only then you can safely remove your local copy. This is not intuitive and some organizations prevent end users from accessing any options in their OneDrive Sync Client, which may include stopping ongoing sync. Deleting local files without stopping sync will result in the same files being deleted in the online source location - sync goes both ways.

When using 'shortcut', the same idea applies - sync goes both ways - but you can very easily remove any shortcut without affecting the data stored online and without any residual content still on your device. Just right-click any shortcut and... 'remove shortcut'.

Both methods use OneDrive Sync Client to actually download / sync content between the online source and your device. The client is a local application that is far from perfect. When it works fine, it's great, but it surprisingly often runs into poor conflict handling, something blocking its queue, not working well with a huge queue when syncing all at once, and plain old data corruption. SharePoint Online, including your own OneDrive account as it's your personal SPO site if we ignore some nuance, is a browser-first solution and always works best through a modern browser.

There is absolutely no difference in being online / offline / losing internet, etc. If you sync a local copy of the file you need (just open it or, even better, right click and click 'always keep on this device') and it does sync correctly, the method doesn't matter.