The first formula is regarding the field created by a small arc length of a possibly-curved conductor. If we integrate that expression over the whole curve of the conductor, then we get the field at a point due to the conductor.
The second expression is the integrated version of the previous formula for an infinitely long straight wire. I wouldn't really consider the second to be the Biot-Savart law; rather it should be the field of a long straight conductor.
As an extra point, I'd like to say that it'd be much easier to find the field of a long straight wire conductor using Ampere's law rather than Biot-Savart.