The battery for a DC or AC UPS can be the same, depending on what you need.
In a DC UPS the load output often effectively comes straight from the battery, e.g. in telecoms or radio comms systems.
This has the advantage of reliability and efficiency as there is no converter between battery and load.
With an AC output UPS some electronics are required (or even a motor and generator!) To generate the AC waveform.
Often the DC voltage cannot practically be high enough to generate the peak AC waveform voltage. Typically in small UPS units of a few kVA a DCDC “boost” stage will produce enough voltage to permit a further stage to generate this into an AC waveform. This is common in “on-line” UPS systems due to efficiency. Alternatively, the low voltage DC is formed into a low voltage AC current and stepped up via a mains frequency transformer. This is common in “off-line” units like the APC Smart-UPS from a few years ago (and possibly still currently).
Higher power units will use high voltage battery banks (lots of series cells) so only AC waveform creation is required. Some systems may still employ a DCDC stage or else just rely on the AC converter stage to control the output voltage.
However it is produced, the conversion from battery DC to AC needs power and results in conversion losses. With poor power factor, an AC UPS efficiency falls due to higher current resulting in higher losses. The battery will still deliver active power + losses due to reactive power I2R losses.
So, I would say the main difference is the battery size. To supply the same energy (power and time) an AC UPS battery needs to be bigger / have a higher capacity to cover the losses in the conversion stages.