An option gives the right but not the obligation to buy (or sell) something at a given price (the "strike") in the future. Equity in this case is that thing. The equity can be structured such that it sits behind other investors in liquidation preferences etc meaning it doesn't have economic value until the share price/valuation hits a particular hurdle (similar to the strike price on an option).
So while they are similar in terms of economic impact, the equity can have other rights attached (eg voting, ROFR for dilution etc) that a founder might want. Also crucially you already have the equity so you don't have to pay up to buy it (as you would in the case of a call option).
Finally as always there may be tax implications which make the difference meaningful. Tax arbitrage is often part of the calculus for any kind of weird compensation package.
So while they are similar in terms of economic impact, the equity can have other rights attached (eg voting, ROFR for dilution etc) that a founder might want. Also crucially you already have the equity so you don't have to pay up to buy it (as you would in the case of a call option).
Finally as always there may be tax implications which make the difference meaningful. Tax arbitrage is often part of the calculus for any kind of weird compensation package.