I have two rose bushes that are close to each other and about 10 years old.
A few months ago when rose bush 1 started to flower I noticed that the buds
were different from the rest of the plant. Few days past and I noticed that
the new buds on rose bush 1 were the same as rose bush 2.
Sure enough when the bud opened the flower was identical to rose bush 2 and
very different from rose bush 1.

Very simple explanation. The grafted top of your rose bush died and what you
see now is the understock blooming. If flowers on rose bush #1 are identical to
#2, then the same thing happened to bush #2 years ago. It is a common
occurrence on older hybrid tea roses which have been planted too shallow and/or
neglected.
If part of the original top of #1 is still alive, prune out the top growth on
the understock completely at once, and you may save it. Consult your nearest
garden center or a good rose book, to prevent the problem in the future. Dig up
#2 and discard it, unless you really like the flowers.
Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)