Some thoughts...
As mentioned by LA, when ordering roses by mail, "bare root" and "grafted" are usually synonymous. The reason an "own root" rose wouldn't be shipped bare root is that the plant is young, has never been in dormancy, and is so small it probably wouldn't survive having the soil taken away from the roots and shipped.
Bear root, grafted plants are usually in their second year, and will usually grow tall and wide in their first year in a garden. The potential risk of grafted roses is that if you live in a cold zone is that sometimes very harsh winters can kill or injure a bud union (planting the graft a few inches below ground, and giving the rose winter protection above ground, can overcome this risk).
On the other hand, own-root roses are typically very small, new plants in their first year. Many of them will not grow very tall and wide in their first or even second year. Some members have had really poor experience with own-root roses, where they never really took off and remained disappointingly small for years, or just faded away. Personally, I've had pretty good experiences with them. Some have done amazingly well their first year, and others took their time to fill out over several seasons. The advantage they offer is that if you live in a cold zone, you don't have to worry about a bud union being harmed during a severe winter. If the canes on an own-root rose completely die back to the ground during a cold winter, new canes will emerge "true to cultivar" from the original roots. And as mentioned, you don't have to worry about suckers coming out from the root stock.
There are also arguments to be made about own-root roses being virus free because they aren't grown on root stock or from cuttings that could be infected, but some of us here don't really care much about rose mosaic virus one way or another, so I won't get into that.
I have about 100 grafted and 25 own-root roses. I specifically buy own-root for three reasons: (1) when I want to plant a rose in an area that gets harsh winter sun and winds; (2) when I want to buy a rose from a nursery during the late spring, summer, or early fall, when other mail-order nurseries aren't selling anything; or (3) when I can only get a particular cultivar from an own-root nursery.
Hope this information helps.
