Is hydraulic fluid for a jack different from hydraulic brake fluid in a car?

Yes, they are different.

Brake fluid will eat the jack seals. Some people might suggest Transmission Fluid or motor oil, but there's really no need -- hydraulic jack oil is inexpensive and easily available at many tool, hardware, and auto parts stores. And it's supposed to keep seals in the best possible condition. Here are a few part numbers you can search for:

GUNK 4KTN1 Hydraulic Jack Oil, 12 Oz

Gold Eagle Heavy Duty Jack Oil - JF16

"Ace" Hydraulic Jack Oil (at Ace Hardware Stores).

As of this writing in mid 2010, a 12 oz bottle is about $4.