The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap.

However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense:

Consider a notional of 1 and a fixed investment horizon $[0,T]$. Ignore second order effects (e.g. daily margining).

• If you buy a variance swap at $t=0$ at a price of 20% (variance strike in volatility units) and that the realised volatility over the contract's life is measured at 25% at $T$, you will lock a profit: $25^2-20^2=225$.
• If you buy a volatility swap at 20% at $t=0$ (volatility strike) and that the realised volatility over the contract's duration is measured at 25% at $T$, your profitt will be: $25-20=5$
• If you enter a VIX future at 20 (variance swap par rate expressed in vol units) at $t=0$ and unwind your position at 25 at $t=T$, you'll have made $25-20=5$.

The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap.

However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense:

Consider a notional of 1 and a fixed investment horizon $[0,T]$. Ignore second order effects (e.g. daily margining).

• If you buy a variance swap at $t=0$ at a price of 20% (variance strike in volatility units) and that the realised volatility over the contract's life ends up being 25%, you will lock a profit: $25^2-20^2=225$.
• If you buy a volatility swap at 20% at $t=0$ (volatility strike) and that the realised volatility over the contract's duration ends up being 25%, your profitt will be: $25-20=5$
• If you enter a VIX future at 20 (variance swap par rate expressed in vol units) at $t=0$ and unwind your position at 25 at $t=T$, you will have made $25-20=5$.

Quantuple and I had a good discussion on this and here is the bottom line: the price/value of the VIX Index itself is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap. However, if you are to trade a delta one contract on the VIX index, the exposure you gain is more comparable to one from the vol swap. For details, please refer to the comment chain below the original question.

The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap. 

However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense:

Consider a notional of 1 and a fixed investment horizon $[0,T]$. Ignore second order effects (e.g. daily margining).

• If you buy a variance swap at $t=0$ at a price of 20% (variance strike in volatility units) and that the realised volatility over the contract's life is measured at 25% at $T$, you will lock a profit: $25^2-20^2=225$.
• If you buy a volatility swap at 20% at $t=0$ (volatility strike) and that the realised volatility over the contract's duration is measured at 25% at $T$, your profitt will be: $25-20=5$
• If you enter a VIX future at 20 (variance swap par rate expressed in vol units) at $t=0$ and unwind your position at 25 at $t=T$, you'll have made $25-20=5$.