1) Beta58 Hypercardioid, SM58 Cardiod - so the Beta58 shows a significantly more directed pickup pattern, which can be good or bad, depending on how much you can control your mic positon and how the monitors are set up (cardioid is least sensitive on the rear, hypercardioid is least sensitive on the sides).
2) Beta58 has a significantly higher output level at similar SPL - if you are working with less expensive preamps this can be a big advantage. With the SM58, it depends a lot on the preamp used if you get pro level results or not.
3) Beta58 has an extended high frequency response, it's high mids / low highs ('presence') are a little faint, but can be easily EQed in, as it takes EQ very well.
4) Overall built quality is better at the Beta58: Hardenend grill, better damping of handling noise. In combination with midrange gear (Focusrite Scarlett series interfaces for example) I prefer the Beta58, as the missing mid / presence peak (it's only 'disadvantge' compare to the SM58) can be EQed in easily, but the higher output level is a real advantage: It gives you a better signal to noise ratio. If you use high end gear with good preamps the SM58 can deliver excellent results, too, though when the budget is big enough you'd rather go with a high-end handheld condenser instead (for example Neumann KMS105), unless you are very loud on stage...