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I think you are searching for the following:
  An exotic {4}-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the boundary case
  An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

The minimal symplectic case.
http://www.msp.warwick.ac.uk/gt/2008/12-02/p019.xhtml

Finally you will perhaps like the following notes by David Gay

This paper will outline in an informal way the construction of a family of 4­manifolds which are homeomorphic but not diffeomorphic.

The first section of the paper (after the introduction, so it is section 2 in the paper), describes the usual construction "of an infinite family of diffeomorphism classes of 4­manifolds in two homeomorphism classes".
  (Roughly speaking, the basic examples of non diffeomorphic but homeomorphic 4-manifolds are constructed as follows : Let $E(1)$ be the algebraic surface, obtained by blowing up 9 points in $\mathbb{C}P^2$. This is an elliptic surface. Let $E(2)$ be the sum of two copies of $E(1)$ (how this is done, is explained in section 2). Define inductively $E(n)$ as the fiber sum of $E(n-1)$ and $E(1)$. By logarithmic transformations you can build from these $E(n)$'s the elliptic surfaces $E(n, m_1,...m_n)$, where $m_1,...,m_n$ are the orders of the transformation. The basic examples of not diffeomorphic but homeomorphic 4-manifolds are such $E(n,p,q)$'s where $p,q$ are relativly prime.)
  Since you asked for compact examples, this doesn't answer your question. Nevertheless I think (hope) that this last link is useful, since it provides a short overview and introduction to non diffeomorphic but homeomorphic 4-manifolds.

I think you are searching for the following: 
An exotic 4-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the boundary case:
An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

The minimal symplectic case:
A symplectic manifold homeomorphic but not diffeomorphic to $\mathbb{CP}^2$#$3 \overline{\mathbb{CP}}^2$ by Scott Baldridge and Paul Kirk

Finally you will perhaps like the following notes by David Gay:

This paper will outline in an informal way the construction of a family of 4-manifolds which are homeomorphic but not diffeomorphic.

The first section of the paper (after the introduction, so it is section 2 in the paper), describes the usual construction "of an infinite family of diffeomorphism classes of 4­manifolds in two homeomorphism classes". 
(Roughly speaking, the basic examples of non diffeomorphic but homeomorphic 4-manifolds are constructed as follows : Let $E(1)$ be the algebraic surface, obtained by blowing up 9 points in $\mathbb{C}P^2$. This is an elliptic surface. Let $E(2)$ be the sum of two copies of $E(1)$ (how this is done, is explained in section 2). Define inductively $E(n)$ as the fiber sum of $E(n-1)$ and $E(1)$. By logarithmic transformations you can build from these $E(n)$'s the elliptic surfaces $E(n, m_1,\dots,m_n)$, where $m_1,\dots,m_n$ are the orders of the transformation. The basic examples of not diffeomorphic but homeomorphic 4-manifolds are such $E(n,p,q)$'s where $p,q$ are relativly prime.) 
Since you asked for compact examples, this doesn't answer your question. Nevertheless I think (hope) that this last link is useful, since it provides a short overview and introduction to non diffeomorphic but homeomorphic 4-manifolds.

I think you are searching for the following:
An exotic {4}-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the oriented case
An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

There exist even a minimal symplectic 4-manifold.
http://www.msp.warwick.ac.uk/gt/2008/12-02/p019.xhtml

Finally you will perhaps like the following notes by David Gay
To clarify this Link:

This paper will outline in an informal way the construction of a family of 4­manifolds which are homeomorphic but not diffeomorphic.

The first section of the paper (afer the introduction, so to be honest, it is section 2), describes the usual construction "of an infinite family of diffeomorphism classes of 4­manifolds in two homeomorphism classes".
(Roughly speaking: $E(1)$ is the blow up 9 points in $\mathbb{C}P^2$, $E(2)$ sum of two copies of $E(1)$, ... From $E(n)$ you can constuct $E(n, m_1,...m_n)$ where $m_1,...,m_n$ are integers (the multiplicities of the fibers over $\mathbb{C}P^1$). The basic examples of not diffeomorphic but homeomorphic 4-manifolds are this $E(n,p,q)$ where $p,q$ are relativly prime. But I think these are not necessarly compact, so perhpas you know this construction)
So see this last link, perhaps as a kind of clarification for the not-compact case or as a kind of short introduction to non diffeomorphic but homeomorphci 4-manifolds.

I think you are searching for the following:
An exotic {4}-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the boundary case
An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

The minimal symplectic case.
http://www.msp.warwick.ac.uk/gt/2008/12-02/p019.xhtml

Finally you will perhaps like the following notes by David Gay

This paper will outline in an informal way the construction of a family of 4­manifolds which are homeomorphic but not diffeomorphic.

The first section of the paper (after the introduction, so it is section 2 in the paper), describes the usual construction "of an infinite family of diffeomorphism classes of 4­manifolds in two homeomorphism classes".
(Roughly speaking, the basic examples of non diffeomorphic but homeomorphic 4-manifolds are constructed as follows : Let $E(1)$ be the algebraic surface, obtained by blowing up 9 points in $\mathbb{C}P^2$. This is an elliptic surface. Let $E(2)$ be the sum of two copies of $E(1)$ (how this is done, is explained in section 2). Define inductively $E(n)$ as the fiber sum of $E(n-1)$ and $E(1)$. By logarithmic transformations you can build from these $E(n)$'s the elliptic surfaces $E(n, m_1,...m_n)$, where $m_1,...,m_n$ are the orders of the transformation. The basic examples of not diffeomorphic but homeomorphic 4-manifolds are such $E(n,p,q)$'s where $p,q$ are relativly prime.)
Since you asked for compact examples, this doesn't answer your question. Nevertheless I think (hope) that this last link is useful, since it provides a short overview and introduction to non diffeomorphic but homeomorphic 4-manifolds.

I think you are searching for the following:
An exotic {4}-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the oriented case
An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

There exist even a minimal symplectic 4-manifold.
http://www.msp.warwick.ac.uk/gt/2008/12-02/p019.xhtml

Finally you will perhaps like the following Lecture notes by Alan Weinstein

I think you are searching for the following:
An exotic {4}-manifold by Selman Akbulut

We construct two compact smooth 4-manifolds $Q_1, Q_2$ which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism $\partial Q_1 \rightarrow \partial Q_2$ can extend to a diffeomorphism $Q_1 \rightarrow Q_2$

Alternatively the oriented case
An exotic orientable 4-manifold by Robert E. Gompf

In the present paper, we exhibit two compact orientable manifolds (with boundary), $M_1$ and $M_2$, which are homeomorphic, but not diffeomorphic.

There exist even a minimal symplectic 4-manifold.
http://www.msp.warwick.ac.uk/gt/2008/12-02/p019.xhtml

Finally you will perhaps like the following notes by David Gay
To clarify this Link:

This paper will outline in an informal way the construction of a family of 4­manifolds which are homeomorphic but not diffeomorphic.

The first section of the paper (afer the introduction, so to be honest, it is section 2), describes the usual construction "of an infinite family of diffeomorphism classes of 4­manifolds in two homeomorphism classes".
(Roughly speaking: $E(1)$ is the blow up 9 points in $\mathbb{C}P^2$, $E(2)$ sum of two copies of $E(1)$, ... From $E(n)$ you can constuct $E(n, m_1,...m_n)$ where $m_1,...,m_n$ are integers (the multiplicities of the fibers over $\mathbb{C}P^1$). The basic examples of not diffeomorphic but homeomorphic 4-manifolds are this $E(n,p,q)$ where $p,q$ are relativly prime. But I think these are not necessarly compact, so perhpas you know this construction)
So see this last link, perhaps as a kind of clarification for the not-compact case or as a kind of short introduction to non diffeomorphic but homeomorphci 4-manifolds.