What is the difference between equating and analogising?
I'm going to assume you're asking this for real, rather than being snippy.
When you equate two things, you are suggesting that they are
functionally the same in some material fashion, that they can be exchanged within a context, and get the same outcome. Mathematically, this is pretty straightforward: 5 = 3+2 means that you can grab 3 of a thing in your left hand and 2 of the thing in your right hand, and you end up with the same number of things as if you'd grabbed 5 things in either your left hand or your right hand or your mouth. Figuratively, it can get a bit more complicated, but for example I might equate my spouse with a grabber tool in the context of getting a light bulb out of the cabinet over the stove. Functionally, either will result in the same outcome: lightbulbs in my hands. Similarly, I might equate a Vespa and a Ford F-150 in the context of getting to the nearest grocery store: functionally, they both result in me arriving at the store so I can get some apple juice.
When you analogize two things, you're NOT suggesting that they have any
functional similarity at all. Analogies are used to illustrate an abstract concept or a relationship, and are always figurative, never literal. Outside of poetry and prose, the main use of analogies is to remove all of the complexities in the item under discussion in order to isolate the specific abstract characteristic or relationship that is the focus of discourse. One might analogize infants to saplings, to illustrate the high rate of growth and change that the baby is undergoing. That doesn't imply that there's any material similarity between saplings and babies - only to isolate the rapid growth as the focus of consideration from all of the many other characteristics that a baby has.
In this case,
@thebeave used an analogy of tax write-offs for start-up costs but not for operations for a small business to the representation of republican opposition to abortion. The abstract concept being isolated is significant support for initiation (launch a new business, deliver a baby) but lack of support for continuation after initiation.