In the United States Senate, if there is a tie in a vote, the vice president breaks that tie. According to the senate.gov website, the vice president cannot vote except when breaking a tie.
Is there any practical difference between this system, and a system where the vice president just gets a single vote like each senator does? If a vote ends up 50 "for" and 50 "against", and the VP breaks the tie by voting "for", is that any different than if the VP just voted normally, and the vote ends up 51 for and 50 against? To put in in another perspective, after the VP breaks the tie and you have a vote of 51 "for" and 50 "against", couldn't any one of those 51 "for" votes be said to be the tie-breaking vote?
In another situation, if the vote is 40 "for" and 60 "against", then the VP doesn't have a tie to break so they don't vote. But even if they did vote normally, it would just be 41 "for" and 60 "against", so the results would be the same.
Is there a situation where there is a meaningful difference between the VP breaking a tie, and the VP getting a vote? Or is it simply a matter of procedure/process; not wasting the VP's time by voting except when a tie-breaker is needed?