You can solve this with the lubridate package.
> library(lubridate)
I don't think /1931 is a common date class. So I'll assume all the entries are character strings.
> RECENT <- data.frame(recent = c("09/08/2005", "11/08/2005"))
> YEAR <- data.frame(year = c("/1931", "/1924"))
First, let's notify R that the recent dates are dates. I'll assume the dates are in month/day/year order, so I use mdy(). If they're in day/month/year order just use dmy().
> RECENT$recent <- mdy(RECENT$recent)
recent
1 2005-09-08
2 2005-11-08
Now, lets turn the years into numbers so we can do some math with them.
> YEAR$year <- as.numeric(substr(YEAR$year, 2, 5))
Now just do the math. year() extracts the year value of the RECENT dates.
> year(RECENT$recent) - YEAR
year
1 74
2 81
p.s. if your year entries are actually full dates, you can get the difference in years with
> YEAR1 <- data.frame(year = mdy("01/08/1931","01/08/1924"))
> as.period(RECENT$recent - YEAR1$year, units = "year")
[1] 74 years and 8 months 81 years and 10 months