母親は子供「を」買い物に行かせました。 と 父親は息子「に」車を洗わせました。 はどう違いますか？説明が難しい場合は、例文を教えて下さい。

洗う is a verb that can take an object, so 息子を洗わせる means "to make (someone) wash his son".
As to 息子を車を洗わせる, it doesn't work as a sentense; there are two objects put in a wrong manner. 息子と車を洗わせる is gramatically OK, but you know it represents a bit weird situation. (A father made (someone) wash his son and his car.)

In contrast, 行く doesn't take an object, so you can say either way, like 子供に買い物に行かせる/子供を買い物に行かせる. But double に feels a bit excessive, and in general, we tend to avoid putting particles successively, so particularly in writing, we might choose the phrase 子供を買い物に行かせました. When we're just talking or chatting, we'd say it either way.

〇父親は息子に車を洗わせました。
×父親は息子を車を洗わせました。
〇母親は子供を買い物に行かせました。
〇母親は子供に買い物に行かせました。

(Actually, there's a subtle difference in nuance between 子供を行かせる and 子供に行かせる, but it's hard to explain in my poor English.
One thing I can say is "to make someone do" is directly translated into "someone「に」doさせる". That is, に is normally used in such a "make someone do" situation, not を. So, 子供に買い物に行かせました does exactly mean "made her son go shopping." and it should sound natural by rights (that is why we use the phrase in talking). But somehow, it sounds a bit weird and excessive, so that's why we write the phrase using を instead of に.

I didn't fully explain about the difference between 子供を行かせた and 子供に行かせた. Sorry for that.)