I use them both – the Blue Dog Ceramic is my favorite speaker especially with Fender amps. I own it in the 12” and the 15” formats – they are still in service. I still have an AlNiCo 12 but rarely use it.
The ceramic has more punch, a rounder sound, higher headroom, tighter bass, more crunch, smoother breakup, woodier with some chime & clear highs. Overall faster response - the sound jumps out even at high volume.
The AlNiCo has sweeter highs, juicier, some jangle, but the bass tends to be flubby when driven hard – won’t be an issue if you play clean most of the time. Noticeable compression at high volume.
I've never had an Alnico Blue Dog, but I have a Celestion Alnico Blue and a Ceramic Blue Dog. The Celestion is in a Vox AC15H1TV and the Blue Dog is in a 5E3 clone. I've mixed and matched the speakers in these amps, and in comparing the the H1TV to a standard AC15CC.
The ceramic Blue Dog has more bottom end... no doubt. No comparison. The ceramic Blue Dog is warmer sounding, the Celestion is more forward and just has scads and scads of that Vox 'chime'.
For my money, I'd go for the Weber.
That's my only contribution. Can't compare the Weber Alnico to the Weber ceramic.