RDS-A is the Netherlands based section of Shell. RDS-B is the UK based section. Because of Dutch law, RDS-A withholds taxes on a portion of its dividend. RDS-B does not. RDS-B generally trades at a slightly higher price because of this.

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The short answer:

The two share classes are essentially equivalent except for withholding tax on dividends.

RDS.A with dividends paid in cash or reinvested via a broker's own DRIP - 15% Dutch withholding tax on dividends.
RDS.A with dividends reinvested via Shell's Scrip Dividend Programme - no Dutch withholding tax on dividends.
RDS.B - no Dutch withholding tax on dividends.

Investors who reinvest dividends via Shell's Scrip Dividend Programme should own A shares (yes, even in an IRA). This is not only because A shares are cheaper, but also because the SDP pays dividends only in A shares. If you are interested in this program, call your broker and ask if they participate and how to sign up.

Investors in tax-deferred or tax-exempt accounts such as IRAs should own B shares unless they are enrolled in the Scrip Dividend Programme. This is because there is no practical way to recover foreign tax withheld in an IRA.

Investors in taxable accounts not enrolled in Shell's SDP can often recover all or a major portion of the Dutch withholding tax via the Foreign Tax Credit. Small investors with total foreign taxes of $300 or less ($600 if married filing jointly) who are not subject to the Alternative Minimum Tax and who have at least $300/$600 in U.S. tax liability get 100% recovery via a single line item on Form 1040; otherwise Form 1116 is usually required and your recovery may be limited. Investors in taxable accounts who can recover all or almost all of their foreign tax withholding should own A shares since they are cheaper. If recovery is significantly limited, B shares may be the better choice. See IRS instructions for the Foreign Tax Credit or your tax advisor for more details.

This post was edited on 1/3/16 at 3:30 pm