When you said function in your original post, I misinterpreted what you meant:
The two expressions
funcPattern[x_] := x^2 + 1
and
function = Function[{x}, x^2 + 1]
have the same behavior to the user, but they are different beasts. (The first one has "DownValues").
Notice that the first is a definition:
?funcPattern
and the second is an assignment.
?function
They are both called "functions" unless one is being very picky. I am not sure what people at WRI call them when they are being picky, but I would say the first "is a held pattern" and the second "is a function"
The difference also becomes apparent when you do this:
function[[2]]
and
funcPattern[[2]]