Difference between duìbùqǐ and duìbuqǐ I have seen examples that show 对不起 as duìbùqǐ and duìbuqǐ in pinyin. For instance:

对不起，我踩到你的脚了。
duìbùqǐ, wǒ cǎi dào nǐ de jiǎole.

对不起，我听不懂。
duìbuqǐ, wǒ tīng bù dǒng.

Similarly, I have also seen the same issue with 没关系 as méiguānxì and méiguānxi.

Why is the neutral tone being marked as a falling tone in these words?
This is only a pronunciation variation in spoken Chinese. Most untrained Chinese people may not even notice the difference between them.

There is no difference between the two variations in your examples.

It's like asking what is the difference between "I'm going to do ..." and "I'm gonna do ...".

Note: There are word that would carry different meaning using neutral tone or not. E.g, 东西 means "east and west" when pronounced as "dōngxī" (without neutral tone), and means "thing" when pronounced as "dōngxi" (with neutral tone).
In a dictionary, a word that must be pronounced with a neutral tone will be explicitly noted with a neutral tone.