Based on Census Bureau's explanation:
"A census block group (BG) is a cluster of census blocks having the same first digit of their four-digit identifying numbers within a census tract. (See also Census Tract.) BGs generally contain between 600 and 3,000 people, with an optimum size of 1,500 people..."
Therefore, the shape/boundary of a block group could be changed between 2000 census and 2010 census depends on the increase/decrease of the total population in that area.
Actually, if you look at the following fact you will realize that there is a significant change in the Census Block group between 2000 and 2010 census:
- 2000 Census: 208,668 Block groups
- 2010 Census: 217,740 Block groups
As a result, I would not suggest you to merge those two datasets together because by doing that can yield inaccuracy and it might not be able to reveal the truth. I strongly suggest you to download and use the most recent data in order to get the most accurate result. (However, if you really wanna merge them together, maybe you can try to join those two datasets by using a field named "GEOID2").
Here is an article provided by Esri, and it gives us a very good example about why accuracy and currency matters in our data:
The map above shows us a fact that the Census Bureau can add new block groups to the inventory but also change the locations and boundaries of existing block groups. Here, Block Group 060730098031 (outlined in blue) shows its boundaries and location in Census 2000; dark red outlines its location in Census 2010 geography. The geocode is identical, but the boundary is completely different. Therefore, performing population or housing unit analyses with current data or projections using Census 2000 geography would yield completely inaccurate results.