I purchased the unlocked model and it does not come with a sim card. I'm on Att, what I've always done it just swap the sim from another phone or go to an Att store and get a new one. In your case, swapping the sim won't be possible because the Note 3 uses a larger sim. Perhaps someone familiar with Verizon will reply with a solid answer but getting a new sim from your carrier should be a simple process.
As for pros and cons, the unlocked version gives you an unmolested version of the software and no carrier bloatware. For me that was the single biggest reason to get the unlocked model. As for cons, in my case the only features or functions I lost by getting the unlocked model was no wifi calling (don't need it) and no native visual voicemail (I use an app to get that function). So I'm very happy with the unlocked model.