Meanwhile refers to time: "She had to leave the house to pick up her daughter. Meanwhile, the burglers broke in and stole her jewelry." At the same time can be used for time but it's more likely to mean however: "We gave him the job but at the same time we were worried if he can do it".

It's a very subtle distinction. Meanwhile you would never use at the end of sentence where at the same time you could. Meanwhile....... It's like when someone says meanwhile it builds up some anticipation of what is to come next. Like in a story you can give an explanation of something happening and then say meanwhile -- meaning here was what was happening WHILE that was happening. When someone says meanwhile often one awaits a longer story or explanation to come. Meanwhile fits certain sentences and feeling tones better. At the same time is a little bit more clinical. One could say it's not as poetic. You did that or this was happening and this other thing was happening at the same time. I can put it at the end of sentence because you don't expect anything more from it. You can say this and at the same time that... but it's not usually as long. You don't expect a story after at the same time. The word meanwhile makes me think of stories and fairy tales. At the same time makes me think more of technical writing. It's subtle and probably not everyone would agree with everything above... but there definitely is a difference between the two expressions. With your high level of English what I would suggest is to google the two words through documents on line to get a better feel of how they are used in actual writing. I think then you'll see what I mean.

I would consider "at the same time" to be the better word in that context. Meanwhile implies event 2 happened during the time it took for event 1 to happen, and ended before event 2 ended. Ex. The older brother studied to enter a top university. Meanwhile, the younger brother became involved in criminal activity.