Doug Smith has a well-known page on the complexities of grading ancient coins, using the Augustus-Agrippa COL NEM dupondius for examples. I'm currently shopping for a "horned Elagabalus" denarius and am experiencing similar struggles. What should I privilege? Take, for example, these two coins currently in the stock of VCoins dealers at comparable prices. I like both coins for different reasons. The top coin has a strong strike and detail but is off-center. The bottom coin is better centered (esp. the obverse) but doesn't have the level of detail the top coin has, at least on the reverse. Of course, one could just keep shopping for the coin that has it all, but sometimes that's the difference between a $65 coin and a $165 coin. So what do you prefer--strike or centering?
I prefer a strong strike as long as the centering isn't to bad, I think your eyes are drawn to stronger detail making the coin overall appeal more interesting.
I'm a strike kind of guy. I haven't collected based on the legends, so I can stomach loss of a legend more than a major design element. Of the two coins, I perfer the above for the better struck reverse and I think overall they style is superior to the bottom.
My evaluation/opinion is somewhat at right angles to the original question. Research on ACSEARCH shows that in March 2018 the following coin sold for $160 (not including buyer's fee): This coin is superior in all respects to both of the OP coins, and surely would be worth stretching one's budget rather than settling for one of the above two, if the coin is important enough in your collection. However, if the horned Elagabalus is a minor addition to your collection, the better struck of the original two also has the important part of the obverse legend (IMP ANTONINUS) and would be an acceptable choice.
I always have the same answer when this question is asked: Neither. You choose when a coin is rare or when price makes all the difference. You will see a hundred of these offered in the next year so you will find the one you want at a price you can tolerate in due time. Ides shows a nice coin but if you don't want to spend $160+, hang in there. Personally, if I were to have just one horned coin, I would rather have the 'other' reverse with a nice clear ELAGAB at reverse right. Mine was $50 at a show ten years ago so it would be more now ---- much more at a big name auction. I bought it to upgrade the facial hair of this one that was $50 also at a show 20 years ago. Strike, centering, style, hair, reverse type, legend legibility ----too many choices. A person could end up with a roll of the things if not careful.
I second Doug's advice. Wait for a better one at auction, for less than either. This cost me only $50 just last year. It's a rare issue to boot (RIC 87):
Here's my favorite Elagabalus aka Marcus Aurelius Antoninus Augustus aka Varius Avitus Bassianus. I don't recall where or for how much.
I prefer coin 1, but the loss of legend on both sides bothers me too much, so ultimately I'm with Doug and Sev. This one was $46 in the most FSR auction. I liked the horn (rare bust variant of his last issue) and the die clash on the reverse. I got this for the ELAGAB reverse legend. $60 last year.
I collect coins for their historical association and the clarity and completeness of their inscriptions. Therefor neither of those coins is particularly appealing to me. Having said all that, I do admire well engraved portraits and depictions with fine detail. But it is difficult to find coins that have it all. Something like this (best example I could find quickly): RIC Volume VI, Lugdunum, No. 17a, Constantius, Caesar of the West ..... and this coin I just bought from Victor Clark (great coin from a great dealer): RIC VI, Alexandria, No. 99b, Constantine as Filius Augustorum
Nice Elagabali (?) everyone. I have but one, but with (as Doug prefers) the "ELAGAB" reverse. It came in an eBay lot last year, $9.33. It has problems, but price is always a big factor in my purchases: Roman Empire - Denarius Elagabalus (221-222 A.D.) IMP ANTONINVS PIVS AVG, laureate, horned, bust right / SACERD DEI SOLIS ELAGAB, Elagabalus sacrificing over lighted altar, star in right field. RSC 246; Sear 7542; RIC 131.