Celeste is a small fig but is the sweetest of the fig family. It has a number of nicknames, a common one is "honey fig". An important different is the Celeste has a "closed" eye, protecting it from insect spoiling. The Brown Turkey is a large fig with an "open" eye and not as sweet. I believe the Brown Turkey is more popular fig than the Celeste.
Both have a similar skin coloring.
Both are cold resistance...but I can testify that the Celeste will burn if not protected from the cold, even in South Texas where winters are typically mild.
As far as physical differences of the trees, I am not sure.
The Celeste leaf is very large with 5 long fingers (see photo)

Celeste Fig leaf photo

My 'Celeste'. Isn't large Brown Turkey actually California, or Thompson's Improved Brown Turkey,or San Piero. I'm referring to English/Eastern Brown Turkey.