Does the difference between random seeds matter?

Will this:

new java.util.Random(/* random seed */ 0)
new java.util.Random(/* random seed */ 1)

result in somehow more "less random" / "more similar" random generators than this?

new java.util.Random(/* random seed */ 0)
new java.util.Random(/* random seed */ 1000)

In other words: do I risk having similar series of ints obtained from the random generators if their random seeds are similar?