Interesting that you should mention those two drives. The DH-20A4P has been my standard DVD scanning drive for many years, ever since I bought my first one. But the SHM-165P6S is still what I consider to be my reference drive.
The short answer is that they are both very good DVD scanners and either would be a good buy. Personally I would try to buy both drives, especially if they are new old stock (or ridiculously cheap secondhand drives in good working order). But if they are secondhand and you can only buy one then I would be inclined to get the DH-20A4P as it is slightly newer. If they are new and you can only buy one, it's a tricky decision - they're both good scanners, slightly different but good. In that scenario I would probably get the SHM-165P6S, but only because I know that the DH-20A4P is such a common drive on the secondhand or nearly new market. The DH-20A4P is probably a better drive if you wish to scan at 8x speed rather than 4x, but it has its downside as well.
Both are also excellent DVD writers. The DH-20A4P probably has the edge when it comes to writing modern discs of mediocre quality, but the SHM-165P6S probably has a slight advantage when it comes to the best quality discs.
The only areas where one drive is definitively better than the other is when it comes to DVD-RAM discs (reading as well as writing) and writing CD-R discs. The SHM-165P6S has a serious bug when it comes to handling DVD-RAM discs, which can render them completely unusable. It also comes from the period when Lite-On drivers were notorious for high jitter when writing CD-R discs. The DH-20A4P has no such problems and is a decent CD writer.
As for software, I still prefer good old CD-DVD Speed. Not only is it free, but it is less buggy than Opti Drive Control. I generally only use Opti Drive control for Blu-Ray scanning or with my Plextor drives as an alternative to PlexTools. Although it is newer than CD-DVD Speed (or Plextools), it doesn't seem to run as well on Windows 8.x or 10 - getting it to scan at the selected speed is a particular problem.
[Optional: If you want the long answer, read on]
The DH-20A4P is a very good drive. It can scan consistently at 8x speed as well as 4x (the traditional standard scanning speed for Lite-On drives) and it stabilises the speed reliably before scanning begins. [Some drives, such as the almost identical LH/DH-20A1 series adjust the speed just as scanning begins which produces elevated errors at the start of discs.]
The only problem with the DH-20A4P is that it drops more samples than I would like. If you look at the statistics window show at the end of the CD-DVD Speed disc quality test it shows the average scanning interval. For a Lite-On DVD scan this should ideally be 1.00 ECC, but my DH-20A4P drives consistently score ~1.20 ECC at 4x speed and >1.30 ECC at 8x speed (indicating that samples have been lost or skipped). This should be perfectly acceptable and nothing to be particularly concerned about, unless you're an old-fashioned CDFreak who known that some of the 60+ other drives in your collection can consistently scan at 1.00-1.02 ECC.
At this point I should emphasise that you are highly unlikely to ever be able to see the difference between a 1.20 ECC scan and a 1.00 ECC scan - the difference would almost certainly be less than you would see by scanning the same disc twice in the same drive. [However it appears to explain why my Lite-On Blu-Ray writers, which show an average scanning interval of ~2.00 instead of the expected 1.00, always reports PIE error levels half that of a Lite-On DVD writer.]
My SHM-165P6S was a drive I absolutely loathed when I first got it, but it received some major firmware updates which transformed it into one of the all-time classics (for DVD±R/RW, but not DVD-RAM or CD-R). It can scan at 4x with a low average scanning interval (typically 1.02, IIRC). At the time 8x scanning was considered too fast for reliable results with Lite-On drives (the results tended to emphasise the reading ability of the drive more than the quality of the disc). But the SHM-165P6S was one the first Lite-On drives which could produce useful scans at 8x speed. My own experience was that on good quality discs the results would be similar to a 4x scan. On mediocre discs the error averages would often be elevated, but it was good enough to be useful at 8x - any suspect discs would be rescanned at 4x, but if the result at 8x looked good I was satisfied that the result could be trusted.
The reason it is still my reference DVD scanner is partly because it was the first drive I had which produced consistent results I could trust and partly because of where it fits in the DVD writer family tree. I class it as Lite-On's first 'second phase' 16x drive - my own designation for the first drives which (in my opinion) could be considered fully developed DVD writers. (The boundary is around late 2005, but the exact date varies by manufacturer - NEC's drives actually went dramatically downhill at this point.) To be a useful scanner (that is one which can reliably allow one to differentiate between a 'good' disc and a 'bad' one) one needs a drive which is not only consistent, but is fairly representative of the reading ability of a typical drive. Newer drives were generally better at reading discs than older models, so would report fewer errors. DVD-ROM drives (especially those in DVD video players) on the other hand are almost invariably worse at reading discs than writers. For me the SHM-165P6S is a good balance. At 4x speed it produces consistent results which are similar to those from later drives, but I am confident that it will allow me to identify discs which are likely to have playback problems on other drives in the future - which after all is the whole point of disc scanning.
But then again my DH-20A4P drives have proven to be equally good in that respect. So I use one of those (the external version) as my main day-to-day DVD scanning drive. The SHM-165P6S is brought out to offer its opinion on the very rare occasion that I get a result that I am doubtful of, or when I want to evaluate some very special discs.
Take you pick (or buy both if you can afford to ), then have hours of fun scanning with your new drive.