What is the difference between CW-complex and Cellular complex? Is every CW-complex is a Cellular space? Is its converse true? 
If it is true then what is the difference between them?

We include the definition of CW-complex in algebraic topology given by Whitehed in 1949:
Definition. A CW complex is a Hausdorff space $X$ together with a partition of $X$ into open cells (varying dimension) that satisfies two additional properties:


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*For each $n$-dimensional open cell $C$ in the partition of $X$, there exists a continuous map $f$ from the $n$-dimensional closed ball to $X$ such that

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*the restriction of $f$ to the interior of the closed ball is a homeomorphism onto the cell $C$, and

*the image of the boundary of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than $n$.  


*A subset of $X$ is closed if and only if it meets the closure of each cell in a closed set.



Definition. A cellular space is a topological space $X$, with a sequence of subspaces 
 $$X^0\subset X^1\subset X^2\subset \cdots \subset X,$$
 such that $X=\bigcup\limits_{n=0} X^n$, with the following properties:


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*CS(1) $X^0$ is a discrete space.

*CS(2)  for each positive integer $n$, there is an index set $A_n$, and  continuous map $\psi_i^n: S^{n-1} \to X^{n-1}$ for each $i\in A_n$ and disjoint copies $D^n_i$ of $D^n$ (one for each $i\in A$) by identifying the points $x$ and $\psi_i^n(x)$ for each $x\in S_i^{n-1}$ and each $i\in A_n$.

*CS(3) A subset $Y$ of $X$ is closed iff $Y\cap X^n$ is closed in $X^n,$ for each $n\geq 0$.

 A: the brief answer is no, every cellular complex is not a CW-complex. The inclusion goes the other way around: CW-complex are cellular complex satisfying two additional properties: 


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*Closure-finiteness $\bf{(C)}$: The closure of each cell is covered by a finite number of cells

*Weak-topology $\bf{(W)}$: a subset $F$ is closed in $X$ if and only if the intersection between $F$ and every cell of $X$ is closed. 


A cellular complex not satisfying $\bf{(C)}$ is for instance the 2-disk made of a 2-cell attached to an infinity of $0$-cell (one for each point of its boundary). 
An example of a cellular complex not satisfying $\bf{(W)}$ is:
$$ \{ 1/n \ \ ;  \ \ n \leq 1 \} \cup \{ 0 \} \subset \mathbb{R} $$. 
To know more: 


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*http://www.profmath.uqam.ca/~powell/Expose-C.W-Complex.pdf

*https://is.muni.cz/el/1431/jaro2015/M8130/um/39015882/at02n.pdf
They both adress this specific question.
A: Both concepts as defined in the question agree. This is proved in the appendix of Hatcher's book "Algebraic Topology". See Proposition A.2.
See also the references given by Jeanne Lefevre.
A: A CW-complex is built inductively, with cells of dimension $n$ only allowed to be attached in the $n$-th step. A cell complex is similar but cells of any dimension may be attached in each step, so there exist cell complexes that are not CW-complexes.
