No... not necessarily. But, maybe... in some cases.

Alan - this is the best all-encompassing analysis of the issue I've EVER read! ... Thanks so very much ....

It depends upon WHAT you shoot and HOW you shoot it.

If you PRINT big... really big... use a lot of wide angle lenses stopped down for great depth of field, and don't do much or any telephoto work.... FF might be ideal for you.

BUT if you shoot sports/action or small wildlife from a distance.... and don't want to invest a small fortune in huge, heavy lenses and hire a Sherpa to help carry them... and generally don't print much larger than 13x19", then a crop sensor camera is probably a better choice for you.

Used to be that FF was a lot better for low light shooting... and to a more limited extent it still is. But improvements in noise handling across all types of sensors and better noise reduction software have made this less of a consideration. Now there's still about a stop difference, where FF is superior for noise handling. Unless it's one of the ultra-high resolution FF cameras, in which case it may be no better or even worse at high ISOs than some of the most recent generation of croppers. For example the full frame, 50MP Canon 5DS-series models maximum setting is ISO 6400, expandable to 12800. In contrast, the APS-C 7D Mark II can be set as high as ISO 16000, expandable to 25600 and 51200. The full frame Canon 5D Mark IV with it's more moderate 30MP sensor (less crowded with tiny pixel sites), can be set as high as ISO 32000 and is expandable to 51200 and 102400. How usable are the extremely high ISOs? Well, that comes down to the user and their intended purposes. Personally I've used 7DII as high as 16000 for moderate size prints (8x10/8x12), and made even larger from ISO 5000 and 6400 images. IMO, the 5DIV is good for about a stop higher ISO. But the 5DS models with their very crowded sensors (pretty close to the same pixel density and pixel site size of 7DII, if memory serves) don't even offer anything beyond 12800.

The Canon 1DX Mark II... which is BOTH full frame AND uses a very modest density 20MP sensor... is probably the "king of high ISO" among Canon DSLRs. It's native range goes to 51200 and it's expandable as high as ISO 409600! (I'll leave it to you to judge if those ultra-high ISOs are actually usable.)

A landscape or architectural shooter might prefer full frame for the great detail that they can capture and that they are usable to smaller f-stops before diffraction becomes an issue, though the difference wouldn't be very apparent until really large prints were made with the images (say 16x20" or bigger). This type shooter also is more likely to be using or able to use a tripod for stability, so is able to make longer exposures and avoid the need for ultra-high ISOs. Full frame also is ideal for wide angle lenses, which are popular for landscape and architecture shots.

But, the corollary is true, too... that a telephoto user can benefit from a crop sensor camera. Basically a sports or wildlife photographer would need to haul around much bigger/heavier and far more expensive telephotos, if they wish to use a full frame camera. For example, on crop-sensor cameras I use an easily hand held, 2.5 lb., 9" long, 3" diameter (77mm filters), 300mm f/4 a lot for sports photography. In order to frame subjects the same way with a FF camera, I'd need to use my 500mm f/4 lens... which weighs about 8 lb., is 15" long, over 6" diameter (150mm front element can't be fitted with filters)... pretty much a "tripod only" lens that makes me a lot less mobile. While I love using the 500mm for some things... when it comes to sports, "No thanks!". Not to mention that the 300mm f/4 lens costs about $1350, while a 500mm f/4 costs around $8000!

Folks also fool themselves into thinking "FF is better", with their post-processing procedures. The mistake they make is evaluating their images "at 100%" on their computer monitors. While it's fine to use 100% or even higher magnification for retouching and other purposes, it makes no sense to evaluate detail, sharpness or even focus accuracy at the level. Think about it.... most computer monitors render roughly 100 pixels per inch. A 24MP DSLR in any format has 4000x6000 pixels. Viewing it "at 100%" is like making a 40" x 60" print, then viewing it from only 18 or 20" away! First, most people don't ever print anywhere near that large (and if they did there are "up-rezzing" programs they'd be likely to use). Second, even if you did print that large you'd likely never view it from so close. By the time the image is finished for printing at the sizes most people actually use, very likely viewing and evaluating it for sharpness, fine detail and focus accuracy would be more sensibly done at 33% or even 25%.

But, what about cropped images? Wouldn't a FF camera's image cropped down to APS-C size be superior? Quite simply.... NO. In fact, it will be worse. Unless a FF camera is extremely high resolution, by the time you crop a 24MP, 30MP, 36MP or even a 42MP image down to APS-C, you actually end up with a lot less resolution than you would have seen just using a crop sensor camera instead. It's been a few years since we tested it... but we put a full frame, 21MP Canon 5D Mark II up against an 18MP Canon 7D (original version).... cropping the FF image down to match the size of the APS-C camera. That's around a 60% crop, so only around 8MP remained of the FF camera's image... and with more than twice the resolution to work with the APS-C camera's image was clearly superior (even though the original 7D uses a strong anti-alias filter that makes for somewhat soft images, prior to post-processing).

Other considerations....

A full frame camera requires full frame-capable lenses, which tend to be bigger, heavier and more expensive. So not only does the camera typically cost more, but the lenses you use upon it often are more expensive too. A crop sensor camera, on the other hand, can use both full frame-capable and crop-only design lenses.... giving you greater selection and potentially some size/weight/cost advantages.

Often crop sensor cameras are faster shooting, too... higher frame rates, higher flash sync speeds, faster buffering and saving of images, etc. And, full frame cameras may be noisier and more prone to internal vibrations, due to the much larger mirror and shutter mechanisms they have to use. (I generally avoid using my FF camera very close around horses... It's louder shutter/mirror slap has caused a few of them to "go vertical"... which ain't fun.) Of course there are exceptions.... Canon's faster frame rate camera is their full frame 1DX Mark II, capable of 14 frames per second. So-called "mirrorless" FF models from Sony also can be quieter and have less internal vibration. Heck, Fujifilm even has a mirrorless medium format camera now... which I haven't used but wouldn't be surprised if it were quieter operating than an APS-C DSLR. We're also seeing some cameras starting to use electronic shutters.... Combined with mirrorless, that can eliminate most moving parts to potentially offer much quieter operation that might be ideal for candid shooting, weddings, courtrooms and more.

For most people and their real-world uses.... a crop sensor camera is a better choice. Personally I use both formats... but I shoot a lot of sports and some wildlife, so my crop sensor cameras see much more use than my FF. I find I use FF about 10% of the time.