Leasing a vehicle in the company name means 100% of the cost can be expensed on the business books, including the lease payments in full, fuel (if used 100% for business), and ongoing maintenance costs. If you purchase the vehicle not only is the accounting for it much more complicated but you don't get the immediate benefit of the tax savings due to rules surrounding the accounting for assets in a company.
There are rules that govern leasing vehicles for the business, however, just as there are for owning vehicles in the business:
You may be asking: "Why not buy though? Isn't it better to own an asset than carry a liability?" The truth is that you always want to buy an asset that appreciates in value and rent/lease assets that depreciate in value. If we're talking about a building or piece of land then buying is better because that asset is likely to go UP in value over its life, or at least has a better chance of that than most other purchases. A vehicle however only ever goes DOWN in value the moment you roll it off the lot (with few exceptions) which makes it likely that in buying a car you will not only carry the asset, unable to take full advantage of the expense of purchasing it, while also carrying a liability (if financed).
Some vehicles can be depreciated 100% in the first year if purchased, which can create a HUGE tax savings impact for your business but these generally apply to "fleet" type vehicles like heavy trucks and SUVs. The IRS allows (in some cases) a full 100% deduction in the year of purchase, however, that would still mean you would carry the loan on your balance sheet to be repaid AND you would have to do the same business vs. personal mile tracking as with the lease. This makes leasing a vehicle in most situations the right option.