How do contact transformations differ from canonical transformations? From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101:

[...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated by the Hamiltonian. The system motion in a finite time interval from $t_0$ to $t$ is represented by a succession of infinitesimal contact transformations which is equivalent to a single finite canonical transformation. [...]

How does the contact transformation differ from the canonical transformation?
 A: In the 2nd (but not the 3rd!) edition of Goldstein, Classical Mechanics, the word contact transformation appears in its index, and there is a 13 line long footnote on p. 382, which (among other things) states

[...] In much of the physics literature the term contact  transformation is used as fully synonomous to canonical transformation, [...]

Concerning canonical transformation, see also this related Phys.SE post.
A: Contact transformations were discovered by Sophus Lie in the 19th century. Within this context an infinitesimal homogeneous (time independent) contact transformation:
$$
\delta q^i = \frac{\partial H}{\partial p_i}\delta t,\qquad \delta p_i = - \frac{\partial H}{\partial q^i}\delta t
$$
is a coordinate transformation that leaves the system of equations:
$$
\Delta =
\begin{vmatrix}
dp_1 ,\dots,dp_n\\
p_1,\dots,p_n\\
dq^1 ,\dots,dq^n
\end{vmatrix} =0,\qquad \sum_ip_idq^i =0
$$
invariant [1]. In this context we can interchange contact with canonical according to Qmechanic's answer. 
In the context of differential geometry, we make a distinction between symplectic transformations on $dim(2n)$ symplectic manifolds and contact transformations on $dim(2n+1)$ contact manifolds. This extends the time independent formulation into an extended phase space (time dependent). [2]
We must now take care on how we use the phrase contact. 
In both symplectic and contact frameworks, we can define a canonical structure, 
$$
\theta = pdq, \qquad \Theta = pdq-Hdt
$$
respectively, that becomes invariant under their respective transformations. 

[1] The infinitesimal contact transformations of mechanics. Sophus Lie. 1889. Translated by D. H. Delphenich.
[2] https://arxiv.org/pdf/1604.08266.pdf, Contact Hamiltonian Mechanics, Alessandro Bravettia, Hans Cruzb, Diego Tapias, 2016
