The second should be: I will abide by the company’s code of conduct. The meanings are different. In the first you are stating that you have an obligation and in the second you are stating that you will follow the code, suggesting that you have the option not to. The use of ‘abide’ in your text is old-fashioned and not common.

They are very different. "Obligation" is imposed from the outside. "Abiding" comes from within. The verb "abide" can be either transitive or intransitive. Using it as a transitive verb, you would say "I will abide the code of conduct". Using it as an intransitive verb, you could say either "I will abide by the code of conduct" or "I will abide with the code of conduct" or "I will abide by following..." (your original sentence).

To be obliged is to feel it more as a requirement or responsibility, without much choice. I think to abide by something could mean this, but could also just be to choose to follow someone's wishes. However, in the text you provided, I think "abiding" means something different. It's more like long lasting.