They should do exactly the same thing, yes (as you mentioned yourself omitting --all does not change anything). If you want only absolutely unique sequences, -m 1 is the way to go.
However as you do not specify any other options, bowtie will by default use a seed length of 28 bp and allow 2 mismatches in the seed, plus allow more mismatches after that. Depending on your read length you might want to chose somewhat more stringent mapping parameters so that -m 1 does not remove too many reads (such as -m 1 -l 36 -n 1 or similar).
Best wishes
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