According to the following diagram, an excited state can dissipate via vibrational relaxation or internal conversion. Aren't they the same thing?
(source: Barltrop and Coyle, Principles of Photochemistry, 1978)
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Vibrational relaxation involves the relaxation of an excited-vibrational state molecule to less energetic vibrational modes through the transfer of vibrational energy (heat). This may be via the molecule's environment (e.g. solvent), known as intermolecular vibrational energy transfer (IET), or through redistribution to other vibrational modes, known as intramolecular vibrational energy transfer (IVR). Importantly, there is no change in electronic state. Both IET and IVR are types of vibrational energy transfer (VET).
Internal conversion, however, involves a change in electronic state, through the coupling of two energy-degenerate vibrational modes (between that electronic state it is in, to some lower electronic state). As such, there is no heat emitted through the act of internal conversion (energy is already conserved during this transition).
As an example, imagine a molecule resides in the first excited singlet state, S$_1$($\nu'$) and in vibrational state $\nu'$. It may undergo VET, from state S$_1(\nu')$ $\rightarrow$ S$_1(\nu)$. The heat given out is the energy different between vibrational states $\nu'$ and $\nu$. It may then undergo internal conversion if the vibrational wavefunctions couple electronic states sufficiently strongly cauing an electronic transition, S$_1(\nu)$ $\rightarrow$ S$_0(\nu)$. The molecule is still likely vibrationally excited, and vibrational energy transfer will allow the molecule to return to its ground vibational state (in its ground electronic state).