What makes this fight interesting beyond the usual what-ifs is the role that both forces played on the frontlines of the Cold War, with both sides seeing deployment in foreign countries as a way of projecting the influence of either super power. But that's a commonality. To get into differences, let's turn to the philosophies each side embraced in arms and armament.
To simplify the matter a great deal, the fighting style and skills of the Green Berets reflect American ingenuity, know-how, and think-on-your-feet problem solving. Their weaponry is built to a slightly higher level of technical sophistication. One of the key roles a Green Beret plays is to train and equip foreign forces friendly to American interests. Just look to their motto, De Oppresso Liber, "to free the oppressed." Proficient (given sufficient briefing) with nearly any firearm going back to the turn of the twentieth century, the Green Beret can train and prepare a local fighting force while supplementing their abilities with his already formidable fighting skills.
On the other hand, the Spetsnaz is outfitted with a set of good generalist weaponry rugged enough to take anywhere and survive (nearly) any set of conditions. He's not as comfortable working as a specialist who adapts the situation to his aims, but he can adapt himself to the situation around him. He knows his weaponry is reliable, and he's just as likely to learn from whomever he's fighting alongside as he is to teach them a tactic from his formidable arsenal.
Part of this difference can be seen in the different philosophies both sides have regarding marksmanship and motion on the battlefield. Our Green Berets claimed that the majority of their training time was spent honing their marksmanship skills, and then learning how to move tactically to in order to make the best use of those skills.
Read between the lines, and their initial strategy becomes clearer: to use coordinated firepower to disable their opponent first, before he can affect them. The best defense is a good offense.
Our Spetsnaz claimed something different: for them, tactical movement and marksmanship were inextricably interlinked. They claimed to have spent an equal amount of time learning to shoot as they did learning to move (in order to dodge, to advance, to gain the terrain advantage, to fall back elegantly, etc.).
Again, read between the lines, their strategy is something like: persevere, react, and control the flow and tempo of the battle around you, and you control the battle.