It's a good question

On one hand you say 'well, these conflicts are only 20 years apart - how can they really be that different?'
On the other hand, I suppose it's a bit like saying 'would WW2 British uniforms do for WW1 British uniforms?' - I mean the two conflicts are only 20-odd years apart, the uniforms are still khaki in both cases, and there's a recognisable evolution from one to the other... But as we know - they are totally different in many ways.
I'm not a uniform expert in the period by any means, but I'd venture perhaps the most obvious difference is that the SYW (in Europe, North America and India) is the end of the true tricorn era. 20 or so years later, the bicorne is in fashion, and the tricorn on its way out.
And as Joroas says, the big soldier's coats with skirts (and latterly turnbacks and facings), which have broadly been in fashion since the latter part of the 1600's and haven't changed very much in that time, have suddenly started to shrink to more elegant garments of a slimmer-fit, almost like cut-down coatees presaging the styles of the Napoleonic era. The cuffs and turnbacks become smaller, although the facings remain, codified as regimental signifiers.
Another noticeable distinction is that the distinctive grenadier's mitre cap, which has held sway for 50 or 60 years, has started to evolve (for the British anyway) into more esoteric forms like the bearskin cap. (Although the Hessians are still sporting 'traditional' mitres in the AWI).
All that said, on the wargames table, from more than a couple of feet away, a redcoat looks very much like a redcoat, so I wouldn't worry too much about it
