The answer would depend upon the context so that we know what definition/use of expanding is being referred to.
1 Expand means to make or become larger or more extensive.
2 in physics it means to undergo a continuous change.
3 it means to give a fuller account of something
4 it means for a person to become more mature, less reserved in their character, nature or behaviour.
4 May be British English only I am not sure. You will have to ask some Americans.
so although they are the same:
"continue to expand" you can say a sentence like this "The universe will continue to expand" That is the theory.
Going to expand: you can say this also
"the Universe is going to continue to expand" you are emphasising it will not stop that is the theory.
For a person:
Going to expand:
"They are going to expand for the rest of their lives" = you think fell the person will continue to mature, although people do not often use the word expand in this scenario, they use the words grow/mature
"they just continue expanding, every time I meet them they have grown more mature"
For an account of something: Referred to as "EXPANDING ON"
Continue to expand"
"Why do you continue to expand on everything I tell you constantly giving 5 minute updates"
"Why do you continue expanding on everything I tell you constantly giving 5 minute updates"
"Will the universe continue expanding?" will it ever stop.
continue to expand:
"Will the universe continue to expand?" will it ever stop.
So basically it is the perspective from which you want to ask the question or view the expansion from that determines which version to use, it doesn't affect the meaning