Lots of benefits: Renting is a great way to save miles/depreciation on your daily driver particularly for long chases. Or, if you need a car when you fly in or part ways with another chase partner mid chase, or just to get a car with capabilities yours may not have (maybe newer, more space, better features, better gas mileage, additional ground clearance, more comfort, AWD, etc).
As Warren said, I also highly suggest using the car as if it was your own daily driver: reasonably and ethically both commuting to the chase and near storms. Don't make the mistake some have of perceiving a rental as a free pass to destroy a car via core punching, driving through washouts, getting stuck in thick mud, abusing the tires or engine, etc. Those things should only ever be done in your own property (if then, but that is for another topic). A willful decision to risk or cause damage to someone else's property is unethical. I am not implying anyone in this discussion is doing that or advocating it, just presenting an opinion that people should refrain from that type of use of the rental car.
As has been pointed out, many of these companies are very aware of rentals for chases and will be quick to ban you if they think that is what you are doing. If you damage a vehicle in a storm or high intensity driving situation, I don't agree insurance (whether personal, credit card, damage waiver) has got you covered in all cases. The fine print of almost every rental contract, insurance policy, and loss damage waiver says that accidental damage is covered, not damage resulting from risk or intent, or prohibited uses. The same is true of conventional insurance, they will fight you if they think you risked the vehicle on purpose. The interpretation of deliberate risk or damage is a grey area.
Avoiding hail is pretty easy and you don't miss a lot unless you insist on seeing every grungy HP tornado. Sometimes I think cautious chasers save insurance money since we are weather aware and know not to drive into hail cores and tornadoes.
For myself, I much prefer to chase in my own car as it has the comforts, capabilities, and custom setup I already need (booster, tablet stands, a place for everything, my large amount of emergency and camping supplies, air mattress, etc), but those things can be transferred into the rental if it is a similar class of car.