Both help to test the acomplishment of some of the CLRM (classical linear regression model) assumptions:

Durbin Watson tests for serial correlation.

Jarque-Bera tests for normallity in the residuals.

Durbin-Watson is a test to detect auto-correlation in a time-series

http://support.sas.com/documentation/cdl/en/etsug/63939/HTML/default/viewer.htm#etsug_autoreg_sect005.htm

 Given a large number of observations, the Jarque-Bera test and

Skewness-Kurtosis test will be the alternatives for testing normality. 

http://cef-cfr.ca/uploads/Reference/sasNORMALITY.pdf

Regards

Please let me know if the following references/sites are useful to you:

1.  What is the difference between test of independence and test of ...

https://stats.stackexchange.com/questions/64438/what-is-the-difference-between-test-of-independence-and-test-of-randomness-in-liDW explicitly tests lag 1 autocorrelation, the runs test is somewhat ... It's a bit like comparing a Jarque Bera test against a Shapiro-Francia test ...

2.  (Misspecification) Tests - Professor Menelaos Karanasos

http://www.mkaranasos.com/ECL6.pdfshould be respecified and perhaps estimated by a different method, depending on the exact ... between residuals two periods apart, third$order serial correlation refers to corre$ ... Testing for first order serial correlation: the Durbin#Watson sta# ..... Bera test (JB) which follows asymptotically a chi$square distribution: JB.

3.  STAT 497 LECTURE NOTE 9

http://users.metu.edu.tr/ceylan/STAT%20497_LN9.pptJarque-Bera Normality Test: Skewness and kurtosis are used for constructing this test ... Durbin-Watson test is for regular regression with independent variables.1.  

Dennis

Dennis Mazur

The Durbin Watson statistic is a number that tests for autocorrelation in the residuals from a statistical regression analysis. The Durbin-Watson statistic is always between 0 and 4. A value of 2 means that there is no autocorrelation in the sample. Values from 0 to less than 2 indicate positive autocorrelation and values from more than 2 to 4 indicate negative autocorrelation. The Jarque-Bera Test, is a test for normality. Normality is one of the assumptions for many statistical tests, like the t test or F test; the Jarque-Bera test is usually run before one of these tests to confirm normality. It is usually used for large data sets, because other normality tests are not reliable when n is large.