Hmm...a lot of varying opinions. Not terribly surprising.
As others have said, it really depends on your main purpose/use for this revolver.
It sounds to me as if you are just looking for a nice S&W revolver to add to your stable, without any definitive role for it to fulfill. So, assuming that my presumption is more-or-less correct...I would offer this:
I have the first variant of the 610, a 5 in. barreled, unfluted cylinder, made around 1990-92. I also have a couple 29/629s. One of them is post-lock. So...a couple considerations. First, if you break out an N frame to show off to people, they're going to expect to be seeing a .44 Magnum. Yes, there are other calibers to be found in an N frame...such as .44 Spl, .45 ACP, .45 Colt, 10 MM, .357 Magnum, and perhaps some others. But...most people will associate the N frame with the .44 Mag.
So...if you pull out an N frame, others might be disappointed to see that it's not a "Dirty Harry" gun. Or, more savvy S&W enthusiasts *might* be pleasantly surprised to see the less common 10 MM.
Also...are you looking to get a currently available model, with the IL? Or will you be wanting to find an earlier pre-lock version. IMO, if you're looking for a real addition to your collection, one that will enhance it, and likely appreciate in value...you'll want to look for a nice pre-lock version, hopefully in the best condition possible, with original box and accessories. In this case, I'm presuming you'll only be shooting it very casually, and not a lot of rounds...so the cost/availability of ammo is not really a large factor. Nor is the capability/utility of the caliber an issue either. So...purely as an addition to a collection, the .44 is probably more iconic.
The first variant of the 610 is also rather sought after, so amongst S&W afficionados, the 5 in. M.610 from the early 90's can be very cool. To be honest, the desirability of any model may be very contingent on the barrel length/dash number. Of course, as the desirability of any given model rises...so does the price. Also, purely as an addition to a collection, early blued models (in pristine condition) may be slightly more desirable, in the eyes of a collector. (Blued is not an option for the 10 MM.) Obviously...opinions will vary.
If, however...you're looking to add a nice S&W, which will also be a useful shooter, for recreational use, target shooting, hunting, and/or self defense...you'll probably be better served by a newer model with the IL. I say that only because they are generally less expensive...I think(?). Of course, you might locate a nice, slightly worn older pre-lock for similar money, which...assuming that it's mechanically sound, would be preferable.
So, as a shooter, as others have said...the choice of caliber will be dependent on your primary purpose, and also to some extent, on whether or not you reload, or are planning on reloading. For target shooting/self defense, I would opt for the 10 MM. For SD, the use of moon clips, and the ability to practice with less expensive .40 S&W ammo is another plus for the 10 MM. For hunting or self defense in the great outdoors, I'd probably go with the .44 Magnum. If you're using commercial ammo, the 10 MM is less expensive, if less common. But, if you buy in quantity online, availability isn't really an issue. If you want to be guaranteed of being able to walk into an average gun shop and finding a box of ammo...obviously .44 Mag is the better choice.
One last comment. As others have mentioned, depending on your actual intended usage, there are other models which you may want to consider. Even in the N frame, there is the gorgeous (and expensive) early Model 27, plenty of relatively affordable and very useful M.28s, the later very nice M.627 with 5 in. barrel and unfluted cylinder, the Model of 1988/1989 M.625 in .45 ACP, and the M.625 in .45 Colt. Moving away from the N frames, the M.586/686 are exceptional revolvers...big enough to tame full house .357 loads, but more "carryable" than M.627s. And the M.19/66...much easier to carry, and while more of a handful and not quite as durable as the larger L & N frames, perfectly adequate for a lifetime of reasonable Magnum usage.
So...I feel as though I've rambled on too long, and not provided much *solid* insight. So, I guess I'll close with the abrupt, no-frills answer which you seemed to be seeking. I *love* my 10 MMs...I have 2 Glock 20s, the 610, and I'd dearly love to find a 1066 or 1076. BUT...the unfortunate truth is that the future of 10 MM is uncertain. The .44 Magnum will obviously be around for a long time. So, if you wish to know that ammunition will remain available...I'd get the .44 Magnum.
Best of luck! Don't rush...enjoy the hunt.