Generally speaking, they can have similar functions, but ので is for objective causes and から is for subjective causes.

First, let's compare apples with apples and oranges with oranges. The first sentence can also be written as: 体の調子が悪い"です”ので、病院へ行きます。 体の調子が悪いですから、病院へ行きます。 In this context, the meaning is the same. Even the level of politeness　(but perhaps the first one is even more polite). But if you compare: 体の調子が悪いので、病院へ行きます。 体の調子が悪いから、病院へ行きます。 Then the first sentence is definitely more polite than the second one. (For example, to your boss, the first sentence would be more appropriate.) Also, note that "から”　has other meaning such as "from". e.g. 私は日本から来ました。（I came from Japan). ”ので” cannot be used here. I hope this helps!