subtract value from previous row by group

With dplyr:

library(dplyr)

data %>%
    group_by(id) %>%
    arrange(date) %>%
    mutate(diff = value - lag(value, default = first(value)))

For clarity you can arrange by date and grouping column (as per comment by lawyer)

data %>%
    group_by(id) %>%
    arrange(date, .by_group = TRUE) %>%
    mutate(diff = value - lag(value, default = first(value)))

or lag with order_by:

data %>%
    group_by(id) %>%
    mutate(diff = value - lag(value, default = first(value), order_by = date))

With data.table:

library(data.table)

dt <- as.data.table(data)
setkey(dt, id, date)
dt[, diff := value - shift(value, fill = first(value)), by = id]