I had the following as a "reply" to Louise, but I'm changing it to an "answer".
Good question, Louise. The tests really are quite different with different purposes. Ancestry does an autosomal DNA test which is the same as Family Finder at FTNDA. The easiest way to look at autosomal is that it is mainly about cousin-finding. It can get pretty technical getting into all the tools of autosomal, but it is about finding not only who is related as cousins, but where they connect.
The YDNA at FTDNA is quite different. While autosomal considers all of a person's ancestor, YDNA is only about the male lineage based on the fact that, in many cultures, the surname is passed from father to son AND at the same time, the Y Chromosome is passed (pretty much) intact from father to son. None of the mixing up that happens with autosomal inheritance.
What that means is that 1) in any close family, only one YDNA test is needed and 2) one test alone without finding a match from another family is pretty meaningless. Where YDNA becomes really powerful is finding out if men of the same surname have a common male ancestor within a genealogical time frame.
Here's my suggestion for your son and nephew:
1) I'm guessing that your son's surname is Halpin and he got that name from his biological father. When he takes a test, it will be to find if he is related to other Halpins. (If not, never mind what I'm about to say.) I see that FTDNA doesn't have a Surname project for Halpin or anything similar so that means there may not be any Halpins to compare with. I see from Forebears there are an estimated 16,000 Haplins in the world, mostly in the US. The top 8 countries with Halpin are typical English speaking (UK, Canada, Australia, NZ.) I'm not seeing anything very definitive about the origin in a quick look. https://forebears.io/surnames/halpin#meaning
. So that's the thing... if you want to study Halpins, that's a start.
2) If your nephew is a Halpin and there isn't a thought that he or your son were adopted (or whatever) then there is no point in getting both tested. On the other hand, if this nephew is from another side of the family, then you are ONLY looking at his male lineage which will (likely) have nothing to do with your son's.
As for the test to take:
The available tests are Y37, Y67, and Y111 referring to the number of markers tested. The more the markers, the more precise the results and the more expensive. However, if you buy a Y111 and you have a match with someone taking Y37, then ONLY the 37 matter. Administrators can order Y12 for screening purposes but wouldn't be useful starting out as you would be.
If your brain isn't full yet, there's more. There are TWO DIFFERENT measures of a YDNA test: Y-STR and S-SNP. What I described above is Y-STR. I won't get into the details of what that means but let's say that is useful for comparing families within a more recent time frame, say 4 to maybe 10 generations or so. Y-SNP is for looking at PRE-HISTORIC lineages. For the most part, it is NOT useful for genealogical study. HOWEVER (before someone hits the reply button too soon), as we get more and more samples, some Y-SNP results are becoming more useful in earlier times.
There is also mtDNA, looking at ONLY female ancestors (of men or women). I'll stop here.