目击 and 目睹 are roughly the same, as they have exactly the same definition "亲眼看到" in 《现代汉语词典》。 You can 目击/ 目睹 a traffic accident. You are a 目击者 of the accident. （This saying is very common in news report）.目睹者 is not as often used, although it is grammatically acceptable. For a sports game, I would say 我 到/在 现场看了那场比赛。 我目睹了那个进球/那个失误。

They are synonym. In some phrases they are interchangeable. Like目睹其事，目击其事。目睹耳闻，目击耳闻。 But they are subtle different. 目击to see in a bit of view time. seen a glimpse of what it was happening in A short time of evidence. 目睹 is seen a duration time of process of events. And because the viewers' mind and aim are different(心理不同）. Some consider themselves as 目击，some considerate themselves as 目睹。目睹only emphasize they have seen this evidence happened but not for 目击as a witness. So there is a saying. 视而无睹则为目击。（关注程度不同）that because viewers don't pay attention on it 目击( not carefully see and glance see) we say 目击。usually phrases 目击证人。 In your scenario A to see a race or game they are 目睹球赛。 B, witness an accident that can use 目击and 目睹。but 目击can be witnessed what happens in a freshness or not during long time. and 目睹 to see what the whole event was happening by theirs own eyes.

"目击"——强调“看到”了某一较短时间发生的事。 "目击"——to "see" something that happened in a relatively short time. "目睹"——强调“看到”了某一较长时间内发生的事。 "目睹"—— to "see" something happening over a longer period of time . “目击证人”和“目睹事情”都是看到的意思。 “目击证人”和“目睹事情”--It's all about seeing. hope it helps.