Translation:Because my stomach hurts, I am going to the restroom.
Yes, they are similar; practically synonyms. There's a subtle (and negligible) difference, in the sense that ので is somewhat more objective in describing a situation or state of being, while から is a bit more subjective. Not a rule though. For example:
この こども は まだ ちいさい ので、[because this child is still small] ようちえん に かよいます [it goes to kindergarten]. More literally, the first part can be broken down like: "As for the child, being that it is still small, ...". This is something rather unavoidable.
あめ が ふって います から、[because it is raining] こうえん に いきません [I am not going to the park]. Here, the fact that it's raining is a reason for me not to go to the park. For someone else that might not be the same, or it could be a reason to go to the cinema instead.