What is the difference between filing a 1040 tax form and a 1040a form?

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They are the same form but the 1040a is a shorter version of 1040. In order for you to file the 1040a you have to meet the following:
1. You only had income from the following sources:
Wages, salaries, tips,
Interest and ordinary dividends,
Capital gain distributions,
Taxable scholarships and fellowship grants,
Pensions, annuities, and IRAs,
Unemployment compensation,
Taxable social security and railroad retirement benefits,
Alaska Permanent Fund dividends,
Jury duty pay.

2. Your taxable income (your income after all allowable tax deductions) is less than $100,000.
3. You do not itemize tax deductions.
4. The only adjustments to income you can claim are jury duty pay to your employer, IRA deduction, student loan interest deduction, or penalty of early withdrawel of savings.
5. The only tax credits you can claim are child tax credit, additional child tax credit, education credits, Earned Income Credit , credit for child and dependent care expenses, credit for the elderly or disabled, retirement savings contributions credit, or federal telephone excise tax credit paid.
6. You did not have an alternative minimum tax adjustment on stock you acquired from the exercise of an incentive stock option plan (see Publication 525).

You can also use Form 1040A if you received advance Earned Income Credit (EIC) payments, dependent care benefits, or employer-provided adoption benefits, or if you owe tax from the recapture of an education credit or the alternative minimum tax.

Reference from: 1040 IRS Forms: Find federal income tax forms for filing tax returns