The correct spelling is "be a dear". They both mean the same and are endearing ways to ask someone to do something for you. The term "be a lamb" is not often used nowadays but "be a dear" is mostly used by the older generation. Both would be used by people that know each other well, like a husband and wife, or a mother and child. Similar expressions (in the UK) are "be a darling" or "be a sweetie" which is what I say to my Daughter when I ask her to do something.

Well, actually, it's "be a dear" (a nice person). Simply, yes, but this is jargon that's normally used only between close family members, and possibly between married couples. Not being a UK native, I would say that 'be a lamb' is much less common than 'be a dear'.

I think that they are the same thing, yes.　But remember, I does sound very old.