So there is a slight difference between these two sentences, and though most translations will end up being the same, the nuance changes what is actually being said.
Let's begin with に:
埼玉から東京に行きます。
In a translation that best captures the nuance, I will say this: I am going from Saitama to Tokyo. It is specifically indicated that Tokyo (as a whole) is designated the final destination.
Now for まで:
埼玉から東京まで行きます。
My translation capturing nuance is this: I am going from Saitama to (someplace in) Tokyo. Here Tokyo is part of the final destination, but it's slightly less specific by implication. It's not the complete final destination.
Which one of them is correct?
They are both correct, and nominally interchangeable. It just depends on whether or not you break the final destination into parts. Tokyo, for example has many regions and zones you can break it down into, and can be appropriately marked with both に and まで. Places like the Tokyo sky tree, the south pole, or the Eiffel tower cannot be broken down into smaller pieces, and can therefore must be indicated by the use of に.
How do you determine whether to use まで or に, under similar situation?
Are there any rules for choosing them under a particular situation?
Usually I'd use に. Note that this can change with speaking style, as people have their own particular style of speech. You are free to do as you wish, but usually, I hear に.
Here's a link to the same source as before that can help you learn more about how に, and まで are similar. They are, for the most part, interchangeable, but nuance does change slightly.
