To do this with BEDOPS bedops --difference and bash:

$ while read A <&3 && read B <&4; do bedops --difference <(echo -e "${A}") <(echo -e "${B}"); done 3<A.bed 4<B.bed

Given:

$ cat A.bed
chr1    1   100
chr2    1   100

$ cat B.bed
chr2    50  70
chr3    1   100

This gives the following result:

$ while read A <&3 && read B <&4; do bedops --difference <(echo -e "${A}") <(echo -e "${B}"); done 3<A.bed 4<B.bed
chr1    1   100
chr2    1   100

To confirm the line-by-line comparison, you could replace bedops --difference with paste:

$ while read A <&3 && read B <&4; do paste <(echo -e "${A}") <(echo -e "${B}"); done 3<A.bed 4<B.bed
chr1    1   100 chr2    50  70
chr2    1   100 chr3    1   100

Note that bedops --difference is directional, i.e. it depends upon the ordering of inputs. For instance:

$ while read A <&3 && read B <&4; do bedops --difference <(echo -e "${B}") <(echo -e "${A}"); done 3<A.bed 4<B.bed
chr2    50  70
chr3    1   100

Of course, to make this work reliably, you will first want to verify that both A.bed and B.bed have the same number of lines before you do any of this. You could, for example, test and compare with wc -l and a numerical comparison test operation in bash.