"What is the most fundamental difference between Reform Judaism and Orthodox Judaism?
"How does this difference then manifest itself in the ways these two respective groups live their lives in response to God?"
The fundamental difference is the approach to Torah and the implications of that approach. The Orthodox believe that it comes directly from God and so cannot be changed. All we can do is "understand" (they wouldn't even say interpret) it, and the right to do so has devolved upon rabbis, descendants of the Pharisees who probably began teaching during the Babylonian Exile. The "authentic" understanding of the Torah is encapsulated in the "halachah," the law (literally, "way"). God is thus the law-giver whose literal words must be obeyed. From this comes the concept of mitzvah, which means "commandment."
For Reform, the Torah is the God-inspired attempt by Hebrews/Israelites/ Jews to understand their surroundings and their relationship with God. While it is a holy document, the Torah is rooted in the past, and we can even sometimes discern the circumstances under which certain sections were written down. Reform thus sees development in Judaism, not just through the biblical period but thereafter as well, so that we can continue the process of helping Judaism evolve by coming to our own understandings. We also recognize that Jews in various places developed varying customs and understandings, again proof to us that Judaism is not and never was monolithic.
When Reform Jews relate to God, they do so on a more personal and less mechanistic level than one would through halachah, though I must add that I am sure that many Orthodox Jews also have a very "personal" relationship with God, and many Reform Jews do feel that God demands certain behavior of them. The fact is, Judaism has never really imposed a "belief" on people, though obviously the halachah system implies a specific understanding of God.
The Eternal One spoke to Moses, saying, "Phinehas, son of Eleazar son of Aaron the priest, has turned back My wrath from the Israelites by displaying among them his passion for Me, so that I did not wipe out the Israelite people in My passion." - Numbers 25:10-11