The difference between At the same time and Together
At the same time is also phrase with the meaning: simultaneously.
Together is also adverb with the meaning: at the same time, in the same place.
Together is also adjective with the meaning: coherent.
check bellow for the other definitions of At the same time and Together
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Together as an adverb:
At the same time, in the same place; in close association or proximity.
Examples:
"We went to school together."
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Together as an adverb:
Into one place; into a single thing; combined.
Examples:
"He put all the parts together."
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Together as an adverb:
In a relationship or partnership, for example a business relationship or a romantic partnership.
Examples:
"Bob and Andy went into business together.  Jenny and Mark have been together since they went on holiday to Mexico."
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Together as an adverb:
Without intermission or interruption; continuously; uninterruptedly.
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Together as an adjective (colloquial):
Coherent; well organized.
Examples:
"He's really together."