First of all, these tests are not specifically intended for determining whether a process is a random walk. But once you take first-differences of a random walk, you should obtain an i.i.d. sequence. Then departures from i.i.d.-ness can be established using the two tests.
The Durbin-Watson test (DW) and the runs test (R) examine different aspects of dependence. DW assesses autocorrelation at lag 1, while R assesses the distribution of the length of runs. (I wonder how you have transformed your data to produce the signs for running R.) Thus DW and R are not exactly the same. (Deriving the relationship between them is perhaps a little involved, so I will not attempt it here.)
The two important things are:
- Under independence, both DW and R should show independence. But under dependence, these tests may or may not capture the dependence; this depends on the type of dependence* (DW and R have power against their specific alternatives but not necessarily against other types of alternatives).
- Due to the multiple testing problem, the significance level should be adjusted accordingly when doing formal inference.
*Wow, so many dependencies in one sentence...