Benefits to the Issuer
Delayed dilution. Issuing new shares dilutes existing shareholders; with a CB, potential dilution is delayed and not gauranteed (i.e. would only occur if bondholders chose to convert their bonds into shares if the conversion price is reached).
Coupon discount. The recent rise in interest rates highlights this point perfectly; versus raising in the straight debt market, by offering bondholders a long term call option on the stock (via the potential ability to convert), issuers are able to benefit from a lower cost of capital. Anecdotally, we estimate the discount is somewhere in the region of 1-2% on average.
Benefits to the Investor
Bondholders get paid to wait for the equity story to play out. Convertible bonds - particularly in the balanced range (40-60% delta) - offer equity participation but crucially with downside protection.
Corporate bond + equity call option = convertible bond. Quasi fixed income/equity exposure, which lends itself to lower correlation with traditional asset classes.
Bondholders are long volatility, so in non-trending markets, benefit from richening of the embedded call option.
Also, because of an asymmetric return profile i.e. unlimited upside but with a quantifiable bond floor, CB investors benefit from convexity and this has a volatility dampening affect on the overall asset class returns. One can get exposure to higher growth sectors such as Biotechnology with much less volatility (higher Sharpe Ratio).