Up Front Costs

Up front costs are the amounts which are due immediately upon entering a lease or purchase agreement.

When you lease a vehicle the up front costs are generally less than they are when buying a car.

The up front costs of leasing a vehicle may include:
Security deposit (which is refunded upon termination of the lease agreement)
First month's payment
Registration

Any applicable taxes
Capitalized cost reduction (a one time payment like a down payment but less) Misc. leasing fees the dealer may have

When buying a car, typically, the up front costs are relatively high when compared to a lease.

The up front costs when buying a vehicle may include:
Down payment (generally a rather large sum.  Often 20%-30% of the total cost of the car)
Any applicable taxes
Registration
Licensing fees
Extended warranty costs
Additional Dealer fees are often charged

Mileage

When leasing a vehicle, the amount of miles is limited by the amount agreed upon in the lease agreement--typically, between 12,000 and 18,000 per year. If you exceed the agreed upon amount of miles, you will be assessed a fee for these extra miles.  The fee is typically between 10 and 15 cents per mile. 

If you anticipate using more than the allotted amount of miles you might consider agreeing upon higher miles per year in the beginning of the lease and increasing your monthly payment, in lieu of paying fees at the end of your lease.

When buying a car, mileage has a direct impact on the cars resale value. Higher than average miles on a car will greatly reduce the trade-in value or the overall worth of the car.

Payments

When leasing a vehicle the monthly payments are usually significantly less than that of buying. The monthly payments are lower because you are not paying for the entire vehicle you are only paying for the amount you use.  Depreciation value, taxes and rent charges (interest payments) are what you will be paying for in your monthly auto lease payments. The depreciation value also includes an allowance for mileage.

When buying a car the monthly payments are usually significantly higher than those of leasing. When you buy a car, your monthly payments are not merely a rental fee as in leasing, they are also a breakdown of the actual cost of the car, as well as, fees and taxes that were added on to the total price. Your down payment also affects the size of your monthly payments. As in leasing, a higher down payment will result in a lower monthly payment.

Simply put: a car is a depreciating asset, and as any accountant, broker or financial advisor will tell you: "Buy what appreciates, lease what depreciates."  When leasing, you are not investing your money in a car which will depreciate significantly over the course of the lease, you are simply paying for your use of it.  You would never invest in a stock from a company who projects significant losses which will continue over the course of it's existence, would you?  That is what you are doing when you buy a car!  So, unless you anticipate using more than 20,000 miles per year or have an overwhelming desire to watch your assets depreciate, leasing is probably the right choice for you.

Termination

When leasing a vehicle, if you wish to terminate early, you may be responsible for contract termination charges.

When buying a vehicle, if you want to end your financing agreement early, you will be required to pay off the balance owed on the car. This will include everything you still owe on the car plus any finance charges.

Source: ALCAB - The Auto Lease Consumer's Advocacy Bureau