I've only been to the the Getty Center but I've read several comments that it looks similar/interchangeable with the High Museum of Art in Atlanta. This is understandable since both are Richard Meier designs in his trademark white. Some information:
High Museum of Art - Wikipedia
High Museum of Art - home
Getty Center - Wikipedia
The J. Paul Getty Museum
The High has its roots in 1905 but the Meier building was built in 1983. The museum doubled its space in 2002 with the addition of structures designed by Renzo Piano. It is located in Midtown Atlanta and as far as I can tell, it is not an encyclopedic art museum.
The Getty Center was opened in 1997 and is also not an encyclopedic museum - that's the role of LACMA, which coincidentally, is still undergoing a Renzo Piano makeover that started in 2004. The Getty center is located on a Brentwood neighborhood hilltop among the Santa Monica Mountains and it overlooks both the L.A. basin and Pacific Ocean. The Getty Center is not to be confused with the Getty Villa in the Pacific Palisades neighborhood of L.A., which is an antiquities museum, housed in a stunning replica Roman villa.
So how do they compare? Some things to consider would include the architecture, gallery space, gardens and grounds, permanent collection, special exhibits, setting and relationship with the city, access, user friendliness, museum shop, community outreach, etc.
I'm sure there are people who have been to both, but comments from those who have gone to only one are welcome as well (especially if I've made any errors).