TITLE: Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$)
QUESTION [95 upvotes]: Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B.
What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is false. I'd appreciate pointers to any tutorial-level discussion that contrasts these operators.
Edit: What I took away from this discussion and the others linked is that logicians make a distinction between $\vdash$ and $\models$, but non-logicians tend to use $\Rightarrow$ for both relations plus a few others. Points go to Trevor for being the first to explain the relevance of completeness and soundness.

REPLY [2 votes]: material conditional $\left(\to\right)$
implication$\left(\Rightarrow\right):$$\quad\to$ is true (perhaps in
an axiom system) in the current interpretation
logical implication / (semantic) logical entailment
$\left(\models\right):$$\quad\to$ is true regardless of interpretation
derivability / syntactic logical entailment $\left(\vdash\right):$$\quad\to$
can be proven true regardless of interpretation

(⊢, ⊨, ⇒ are metalanguage symbols, while → is in the object language.)
For example, these two claims are simultaneously plausible: \begin{align}&\forall x\;\; x=x &\Rightarrow &&\forall x\,\forall y\;\;\;  x^2 -y^2 = (x+y)(x-y),\\&\forall x\;\; x=x &\not\models &&\forall x\,\forall y\;\;\;  x^2 -y^2 = (x+y)(x-y) .\end{align}
P.S. Symbolic logic is an area rife with conflicting notation, terminology and even notions; my understanding is eclectically evolving.
P.P.S. To be clear: although I distinguish analytical and synthetic implication ⇒ from logical entailment ⊨, in practice I do frequently use ⇒ (which is better recognised) even when I specifically mean the latter.