Which of them incurs greater sin, the one who breaks his fast secretly in Ramadan or the one who openly breaks the fast?

Praise be to Allah.

Firstly:

It is haram to break the fast during Ramadan for someone who has no excuse such as travelling or sickness, and the one who does that is committing a grave major sin, because of the report narrated by Ibn Khuzaymah (1986) and Ibn Hibban (7491) from Abu Umamah al-Bahili who said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “Whilst I was sleeping, two men came to me and took hold of my upper arm, and brought me to a rugged mountain. They said: ‘Climb up.’ I said: ‘I cannot do it.’ They said: ‘We will make it easy for you.’ So I climbed up until I was at the top of the mountain. There I heard loud voices. I said: ‘What are these voices?’ They said: ‘This is the howling of the people of Hell.’ Then I was taken and I saw people suspended by their ankles, with the corners of their mouths torn and pouring with blood. I said: ‘Who are these?’ They said: ‘These are the people who broke their fast before it was time to do so.’” Classed as sahih by al-Albani in Sahih Mawarid az-Zam‘an (1509). 

If this applies to one who broke the fast during the day, then the one who started the day not fasting and did not intend to fast in the first place has committed a more grievous offence.

The one who breaks the fast secretly and the one who openly does not fast are equal in that regard, because both of them have committed a grievous action and have committed one of the worst of major sins.

Al-Hafiz adh-Dhahabi (may Allah have mercy on him) said: It is firmly established among the believers that the one who does not fast in Ramadan, without being sick or having any excuse, is worse than the adulterer, the collector of unjust taxes and the one who is addicted to alcohol. In fact they doubt whether he is even a Muslim, and they think that he is a heretic and immoral person."(Al-Kaba’ir  157).

Secondly:

The one who openly does not fast incurs greater sin than the one who breaks the fast secretly, because by breaking the fast openly he also shows not respect to the fast, encourages others to break the fast, and spreads evil among the people.

So the one who breaks the fast openly is adding other sins to his sin. Hence the one who conceals his action is more likely to be pardoned than the one who commits it openly, as al-Bukhari (6069) and Muslim (2744) narrated that Abu Hurayrah said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “All of my ummah may be forgiven except those who commit sin openly, and it is part of committing sin openly for a man to do something at night, then in the morning, when his Lord has concealed him, he says: O So and so, I did such and such last night, when his Lord had concealed him all night, but in the morning he discloses that which Allah had concealed for him.”

Al-Minnawi (may Allah have mercy on him) said: “All of my ummah may be forgiven” and may be spared Allah’s punishment for their sin, so if a person conceals it in this world, it may be concealed in the hereafter and he may be pardoned for it."(At-Tanwir Sharh al-Jami‘ as-Saghir  8/160).

It says in al-Bahr al-Muhit ath-Thajjaj (42/761): We also learn from it that it is abhorrent to commit sins openly, and that Allah, may He be exalted, will not forgive the ones who do that, because they openly challenge Him by doing so, and think of those sins as insignificant. End quote.

And Allah knows best.