PHE has been counting all deaths in people who have laboratory-confirmed infection – this is technically robust because it does not require a judgement to be made about cause of death. ONS reports deaths where a doctor suspects COVID-19 as a cause – these data include a clinical assessment as recommended by WHO but are subject to variation in clinical judgement as to the cause of death.
These are two different measures with different strengths and weaknesses. The PHE data series is also available daily, making it more useful for real-time surveillance, whereas the ONS survey only appears once a week and is delayed.