These are 2 different systems.
1) The Air Pump provides oxygen to the exhaust manifold for a more efficient burn of exhaust gasses. The airpump is a belt driven accessory (looks like an alternator, or A/C compressor) that pumps air through hoses and finally into holes in the exhaust manifold (as many holes as cylinders). If the pump shares a belt with any other acessories, then just simply measure for a new shorter belt if you remove it.

2) The EGR, Exhaust Gas Recirculation valve does just that, it re-routes exhaust gasses from the exhaust manifold to the intake manifold. I don't think that you'll see a huge performance difference by taking it off, but it sure does look better, and its one less vaccum operated mechanism to leak and give you a rough idle.

I of course do not endorse the removal of any emmisions related equipment required by law, unless the vehicle if for off road use only.

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Thanks,
Dave

1998 BMW 540i Sport
1995 E320
1992 400E
1989 BMW M3
1979 Triumph TR7 (1st car)
1986 BMW 735i (Sold)
1980 BMW 320i (Sold)