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Are virtual particles and decaying particles the same?
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Decaying particles and resonances are used synonymously in the 
literature; they are complementary views of the same unstable state.

A very sharp resonance has a long lifetime relative to a scattering
event, hence behaves like a particle in scattering. It is regarded 
as a real object if it lives long enough that its trace in a wire 
chamber is detectable, or if its decay products are detectable at 
places significantly different from the place where it was created. 

On the other hand, a very broad resonance has a very short lifetime 
and cannot be differentiated well from the scattering event producing 
it; so the idealization defining the scattering event is no longer 
valid, and one would not regard the resonance as a particle. 

Of course, there is an intermediate grey regime where different people 
apply different judgment. This can be seen, e.g., in discussions 
concerning the tables of the Particle Data Group.

The only difference between a short-living particle and a stable 
particle is the fact that the stable particle has a real rest mass, 
while the mass m of the resonance has a small imaginary part. 
Note that states with complex masses can be handled well in a rigged 
Hilbert space (= Gelfand triple) formulation of quantum mechanics.
Resonances appear as so-called Siegert (or Gamov) states. 
A good reference on resonances (not well covered in textbooks) is
    V.I. Kukulin et al.,
    Theory of Resonances,
    Kluwer, Dordrecht 1989.
For rigged Hilbert spaces (treated in Appendix A of Kukulin), see also
quant-ph/9805063 and for its functional analysis ramifications,
    K. Maurin,
    General Eigenfunction Expansions and Unitary Representations of
    Topological Groups,
    PWN Polish Sci. Publ., Warsaw 1968.


But a very short-living particle is not the same as a virtual 
particle. Often it is a complicated, nearly bound state of other 
particles. On the other hand, virtual particles are essentally always 
elementary. (There are exceptions when deriving Bethe-Salpeter equations
and the like for the approximate calculations of bound states and 
resonances, where one creates an effective theory in which the latter
are treated as elementary.)

Conceptually, an unstable elementary particle is clearly distinguished 
from a virtual particle. In perturbation theory, unstable elementary 
particles are modelled exactly like stable particles, 
namely as external lines in a Feynman diagram. 
Virtual particles in Feynman diagrams are exactly those parts 
of the diagram which are not given by external lines.
In particular, what is real and what is virtual is not affected 
by a diagram rotation - this only affects what is input 
and what is output. 

The difference can also be seen in the mathematical representation.
In an effective theory where the resonance (e.g., the neutron or a 
meson) is regarded as an elementary object, the resonance appears 
in in/out states as a real particle, with complex on shell momentum 
satisfying p^2=m^2 with a complex mass, but in internal Feynman 
diagrams as a virtual particle with real mass, almost always off-shell, 
i.e., violating this equation.

There are also some unstable elementary particles like the weak
gauge bosons. Usually, one observes a 4-fermion interaction and the 
gauge bosons are virtual. But at high energy = very short scales,
one can in principle observe the gauge bosons and make them real.
This means that they now appear as external lines in the corresponding
perturbative calculations, which displays their nonvirtual nature.

In any case, from a mathematical point of view, one must choose the
framework. Either one works in a Hilbert space, then masses are real
and there are no unstable particles (since these 'are' poles on the 
so-called 'unphysical' sheet); in this case, there are no asymptotic 
gauge bosons and all are therefore virtual. 
Or one works in a rigged Hilbert space and deforms the inner product; 
this makes part of the 'unphysical' sheet visible; then the gauge 
bosons have complex masses and there exist unstable particles 
corresponding to in/out gauge bosons which are real. 
The modeling framework therefore decides which language is appropriate.