I would like to show an alternative way for doing such kind of things even often I have the feeling it is not appreciated doing this in that way: using sql.
sqldf(paste("SELECT a.ID,a.Score"
            ,"      , a.Score - (SELECT b.Score"
            ,"                   FROM df b"
            ,"                   WHERE b.ID < a.ID"
            ,"                   ORDER BY b.ID DESC"
            ,"                   ) diff"
            ," FROM df a"
            )
      )
The code seems complicated but it is not and it has some advantage, as you can see at the results:
    ID Score diff
 1   1    40 <NA>
 2   2    36 -4.0
 3   3    32 -4.0
 4   4    28 -4.0
 5   5    24 -4.0
 6   6    20 -4.0
 7   7    16 -4.0
 8   8    12 -4.0
 9   9     8 -4.0
 10 10     4 -4.0
One advantage is that you use the original dataframe (without converting into other classes) and you get a data frame (put it in res <- ....). Another advantage is that you have still all rows. And the third advantage is that you can easily consider grouping factors. For example:
df2 <- data.frame(ID=1:10,grp=rep(c("v","w"), each=5),Score=4*10:1)
sqldf(paste("SELECT a.ID,a.grp,a.Score"
            ,"      , a.Score - (SELECT b.Score"
            ,"                   FROM df2 b"
            ,"                   WHERE b.ID < a.ID"
            ,"                         AND a.grp = b.grp"
            ,"                   ORDER BY b.ID DESC"
            ,"                   ) diff"
     ," FROM df2 a"
     )
)
   ID grp Score diff
1   1   v    40 <NA>
2   2   v    36 -4.0
3   3   v    32 -4.0
4   4   v    28 -4.0
5   5   v    24 -4.0
6   6   w    20 <NA>
7   7   w    16 -4.0
8   8   w    12 -4.0
9   9   w     8 -4.0
10 10   w     4 -4.0