With horizontal scaling:
1. Spin up the new node(s)
With vertical scaling:
1. Spin down the node¹ (hopefully you've got more than one!)
2. Resize it
3. Spin the node back up
4. GOTO 1 unless all nodes are scaled up
OR (somewhat simpler, but at this point you might as well just horizontally scale):
1. Spin up the replacement node(s)
2. Spin down the old node(s)
That is: vertical scaling has more steps, even when done in a way similar to horizontal scaling. Sometimes it's necessary to scale vertically instead of horizontally, though (but in that case you might as well just horizontally scale with bigger nodes).
¹ It's theoretically possible to add and remove resources to/from a running system; some mainframes offer it as a feature (Multics programmers in particular were known to yank hardware out of production systems during off-hours and plug 'em into dev systems for development/testing, then plug 'em back into production for on-hours, all without bringing down the system), and Linux supports hot-adding and hot-removing CPU and RAM provided the underlying hardware supports it (Windows supports hot-adding as of Windows Server 2008 Datacenter Edition, but not hot-removing, last I checked). However, I'm not aware of any PaaS/IaaS providers offering compute instances with this capability; EC2 instances definitely don't support it, last I checked (switching from one instance type to another absolutely requires stopping and restarting EBS-backed instances, and instance-storage instances have to be outright imaged and then reimaged onto a new instance of the desired type).