Short answer: Nothing. They both instantiate a new system.collections.arraylist object.
Details. The New-Object cmdlet “under the hood” calls a constructor method for the type specified plus the arguments provided (in this case none). New() is a default constructor method for system.collections.arraylist, so I suspect that is what New-Object is actually calling.
I wasn’t considering speed difference in your question. The best way to determine that is to use Measure-Command. It will tell you definitively which is faster based on your data set.
Based on my personal experience, it is the same thing, ::new is the original call and New-Object is simply a convenient and straight-forward command to instantiate it.