JPEG compression is standardized. The only part of the algorithm that is configurable is the quantization matrix which is essentially a control of what information to throw away and how much. So unless you are throwing away a huge amount of information (really shitty image), there is no way a different library will somehow save you a bunch of space.

It depends a lot on what's being encoded, which is also why different people (who've actually tested it with some sample images) give slightly different answers. On "average" photos, there's broadly agreement that WebP and MozJpeg are close. Some will say WebP is a little better, some will say they're even, some will say MozJpeg is still a little better. Seems to mostly come down to the samples tested, what metric is used for performance, etc.

I (re)compress a lot of digital art, and WebP does really well most of the time there. Its compression artifacts are (subjectively) less perceptible at the level of quality I compress at (fairly high quality settings), and it can typically achieve slightly-moderately better compression than MozJpeg in doing so as well. Based on my results, it seems to come down to being able to optimize for low-complexity areas of the image much more efficiently, such as a flatly/ evenly shaded area (which doesn't happen in a photo).

One thing WebP really struggles with by comparison is the opposite: grainy or noisy images, which I believe is a big factor in why different sets of images seems to produce different results favoring either WebP or JPEG. Take this (PNG) digital artwork as an extreme example: https://www.pixiv.net/en/artworks/111638638

• Original: 6,080,273 bytes
• MozJpeg (Q88, 4:2:0): 438,687 bytes
• WebP (Picture preset, Q90): 595,634 bytes

This image has had a lot of grain added to it, and so both encoders end up with a much higher file size than typical for digital artwork at this resolution. But if I put a light denoiser on there to reduce the grain, look at how the two encoders scale:

• MozJpeg (light denoise, Q88, 4:2:0): 394,491 bytes (~10% reduction)
• WebP (light denoise, Picture preset, Q90): 424,612 bytes (~29% reduction)

Subjectively I have a preference for the visual tradeoffs on the WebP version of this image. I think the minor loss of details (e.g., in her eyes) is less noticeable than the JPEG version's worse preservation of the grain and more obvious "JPEG compression" artifacts around the edges of things (e.g., the strand of hair on her cheek).

And you might say "fair enough it's the bigger image", but now let's take more typical digital art that hasn't been doused in artificial grain (and was uploaded as a PNG): https://www.pixiv.net/en/artworks/112049434

• Original: 2,184,811 bytes
• MozJpeg (Q88, 4:2:0): 200,933 bytes
• WebP (Picture preset, Q90): 164,078 bytes

Subjectively I once again prefer the tradeoffs made by WebP. Its most obvious downside in this sample is ~~on the small red-tinted particles coming off of the sparkler being less defined,~~ [see second edit notes] probably the slightly blockier background gradient, but I find this to be less problematic than e.g., the fuzz around all of the shooting star trails.. and all of the aforementioned particles.

Across dozens of digital art samples I tested on, this paradigm of "WebP outperforms for non-grainy images, but does comparable or worse for grainy images" has held up. So yeah, depends on what you're trying to compress! I imagine grain/noise and image complexity would scale in a similar way for photos, hence some of (much of?) the variance in people's results when comparing the two formats with photos.

Edit: just to showcase the other end of the spectrum, namely no-grain, low complexity images, here's a good example that isn't so undetailed that it might feel contrived (the lines are still using textured [digital] brushes): https://www.pixiv.net/en/artworks/112404351

• Original: 1,804,869 bytes
• MozJpeg (Q88, 4:2:0): 339,182 bytes
• WebP (Picture preset, Q90): 244,224 bytes

I quite strongly prefer the WebP version here, even though the JPEG is 39% larger!

Edit2: I've corrected the example with the sparkler - I wrote the crossed out section from memory from when I did this comparison for my own purposes, but when I was doing that I was also testing MozJpeg without chroma subsampling (4:4:4 - better color detail). With chroma subsampling set to 4:2:0, improved definition of the sparkler particles doesn't really apply anymore and is certainly no longer the "most obvious" difference to the WebP image!

You must be talking about a kind of jpg I haven't heard of then

Wow. I didn't know this. Are there GUIs to convert images with mozjpeg?