Not in general, depends on interest rates & others' bets for or against.
Shorting means selling without owning (borrowing) the underlying stock, and paying a borrow fee for as long as you hold the position.
Buying (being 'long') a put means paying money up front (the premium) for the option to sell a stock (doesn't matter if you own it or not at the moment) before a certain expiry date.
Typically you would be more likely to hold a short position longer than you'd roll over options, as far as I understand. But it's not clear cut and generalisable because the fee structure is different.
Also buying puts you have no risk if the stock rises - you're out the premium but you were anyway. If you sold short, it's getting more and more expensive to cover if you want to give up on the bet (and borrow fee probaby rising too, so also increasingly expensive to hold).
The VIX index, a mean-reverting proxy for risk aversion on the S&P500, is currently relatively low. This makes buying options, both puts or calls, relatively cheap. Usually volatility is negatively correlated to spot price of the underlying instrument (in this case the S&P500), but if the market remains flat and there's a significant spike in volatility, the position can still be P&L positive.
As for the convexity, assuming the puts he's buying are out of the money, his total position increases in $ value as it becomes in the money and vice versa. The opposite is true with an outright short.
Of the two possible ways to bet against the market we are discussing Burry chose the one with less immediate downside if you have to wait to be right. But there isn't much difference.