In common (spoken) English, both *could* be heard depending on the speaker.
These two mean the same but when a speaker says these two sentences, emphasis could be put on certain words to stress the necessity of getting the work done.
In example number two, I would say:
I have got to catch up with some work. (got being sound more emphatically, would mean, "I am really behind, I have to do my work"
In the first example, as a speaker, I would say
I've got some work to catch up with. (I italicized 'I have' because in this case I would not normally pronounce each word).
This example also *tends* to me more informational and does not stress any urgency.
I am not a teacher -- just a native :)
Student or Learner