The original unit will operate in either constant-current or constant-voltage mode, according to the connected load.
CV: For loads taking 0-300mA at 24V, the voltage will be constant at 24V.
CC: For loads dropping 0-21V at 350mA, the current will be constant at 350mA.
These regimes occur with two separate ranges of load. We shouldn't really talk about the 'resistance' of LEDs since they don't behave like resistors (their current and voltage don't vary in proportion). Instead, let's consider the load on the driver to be a variable resistor, set initially at say 10 ohms, that we are going to increase steadily.
The driver will output a constant 350mA into the load developing 0.35x10=3.5V, and will continue to do so as the resistance is increased, until it reaches 60 ohms at which point the voltage is 0.35x60=21V. This is the highest voltage at which the current is regulated at 350mA and if it increases further, we leave the constant-current regime and regulation is not guaranteed. The limit of 21V is called the 'voltage compliance' in CC mode.
As the resistance increases further and reaches 80 ohms we enter the constant-voltage regime. The voltage stops rising and becomes regulated at 24V, with a current of 24/80=300mA. As the resistance continues to increase, the voltage remains at 24V as the current drops towards zero.
The new driver does not show a constant-voltage mode officially, but if it does go into constant-voltage mode at low load current, it would deliver 34V (not 24V like the old one) and would over-drive a small 24V CV load. OTOH if the load is a 350mA CC one, then because the 34V compliance is wider than the original 24V, it will work just fine as a replacement.