The clarinet is capable of going much lower than the alto saxophone. The alto saxophone does not have the capability of playing most of the written pitches in the lowest register of the clarinet — the chalumeau register. However, the clarinet and the alto sax both have the ability to play notes in the center of the range known as the clarion register on the clarinet. The chalumeau register starts on E below middle C and ends on B-flat below middle C. The lowest note on the alto saxophone is an A below middle C.
In the first octave of the alto sax from A below middle C to D-flat one octave above middle C, the fingerings for alto sax and clarinet do not match up at all. You will need to learn a completely new set of fingerings for this register if you would like to learn to play the clarinet and alto saxophone. The best way to learn new fingerings is to consult a fingering chart and learn your major and minor scales.
The middle register from D an octave above middle C to the B two octaves above middle C use the same fingerings as the clarinet. The fingerings change for the remainder of the middle register on alto sax, starting with C two octaves above middle C. From that point on in the middle register, you will have to learn new fingerings to be able to play both instruments. The middle register is the register that most beginners will play, which makes it easier to switch between both instruments.
Both clarinets and saxophones come in four varieties. While each instrument will sound in a different register, the actual written pitches for all of the saxes use the same fingerings and the same is true of the clarinets. For example, the soprano sax actually sounds a whole step lower than the alto saxophone. But it is given the name soprano sax because of its shrill timbre. The same is true of the clarinets. This makes it possible to easily switch between any clarinet type, whether it is soprano, alto, bass or contrabass. For the saxophones, there are soprano, alto, tenor and baritone saxes that all share the same fingerings.