I want to add some details to Farid Cher's answer as this is a very common problem. Using amatch can do wonders, but with these Spatial objects you should not use base::merge and not access the @data slot. That would inevitably leads to a terrible mess (base::merge changes the order of records, and they would no longer match geometries).

Instead, use the sp::merge method, by using the SpatialPolygonsDataFrame as first argument in merge. Also note the potential problem of having duplicated records. And I added data so that the example is self-contained and reproducible.

library(raster)
#example data.frame
name <- c("Aragatsotn", "Ararat", "Armavir", "Gaghark'unik'", "Kotayk", "Lorri", "Shirak", "Syunik'", "Tavush", "Vayots' Dzor", "Yerevan City","Aragatsotn")
value <- runif(12, 0.0, 1.0)
df <- data.frame(name, value)

# example SpatialPolygonsDataFrame
arm <- getData('GADM', country='ARM', level=1)[, c('NAME_1')]

This

merge(arm, df, by.x='NAME_1', by.y='name')

fails with message

#Error in .local(x, y, ...) : non-unique matches detected

Because there are two records for "Aragatsotn" in df. You could do

merge(arm, df, by.x='NAME_1', by.y='name', duplicateGeoms=TRUE)

But normally the same approach is to use someting like

df <- aggregate(df[, 'value', drop=FALSE], df[, 'name', drop=FALSE], mean)
m <- merge(arm, df, by.x='NAME_1', by.y='name')
data.frame(m)

data.frame(m)
#        NAME_1       value
#1   Aragatsotn 0.421576186
#2       Ararat 0.003138734
#3      Armavir 0.703402672
#4       Erevan          NA
#5  Gegharkunik          NA
#6       Kotayk 0.926883799
#7         Lori          NA
#8       Shirak 0.430585540
#9       Syunik          NA
#10      Tavush 0.121784395
#11 Vayots Dzor          NA

Now, merge does not work well in this case because the names do not match. So you can use

i <- amatch(df$name, arm$NAME_1, maxDist = 3)
df$match[!is.na(i)] <- arm$NAME_1[i[!is.na(i)]]
df
#            name       value       match
#1     Aragatsotn 0.421576186  Aragatsotn
#2         Ararat 0.003138734      Ararat
#3        Armavir 0.703402672     Armavir
#4  Gaghark'unik' 0.682169824 Gegharkunik
#5         Kotayk 0.926883799      Kotayk
#6          Lorri 0.128894086        Lori
#7         Shirak 0.430585540      Shirak
#8        Syunik' 0.163562936      Syunik
#9         Tavush 0.121784395      Tavush
#10  Vayots' Dzor 0.383439033 Vayots Dzor
#11  Yerevan City 0.168033419        <NA>

Almost there, but "Yerevan City" did not match with "Erevan". In this case you can increase maxDist

i <- amatch(df$name, arm$NAME_1, maxDist = 10)
df$match[!is.na(i)] <- arm$NAME_1[i[!is.na(i)]]

But increasing maxDist will not always work. Sometimes variant names are very different. So in many cases you will end up doing some manual replacements like:

df[df$name=="Yerevan City", 'match'] <- "Erevan"

In any case you will want to check if sum(table(i) > 1) == 0; although merge should fail anyway if there are duplicate matches.