I'm not Christian, myself, but this is an intriguing question.
As russianspy1234 has done, I'd like to try to add a little more historical perspective: at the time of the 10 (or, as pointed out, 613) Commandments, the monotheistic G-D wasn't the only game in town. Hence, commandments 1+2 deal with the consolidation of the 12 tribes under one common theological banner. In a very Jewish way, the law is set down simply, but with plenty of space for millions of words of commentary.
The environment was one of chaos; anarchy; limitless (often genocidal) warfare between warlord monarchs; all informed by an unwanted form of tribal pluralism. Heck, there weren't even any 'Jews' at the time: the 12 tribes of Israel weren't even yet united internally, never mind taken together.
In such an environment, it seems sensible to lay down the law, and enforce it by might. The legalistic approach is shown by what happened directly after Moses came down from Sinai. Spoiler: it's not pretty.
Fast-forward a few centuries.
At the presumed time of Christ, Israel was a thing. One could reasonably identify as 'Jewish, from Canaan,' as opposed to 'Canaanite, allied to other tribes.' Roman occupation had managed, for better or for worse, to bring the stability and order to the region through a version of legalism that was somewhat different from the native one; the crisis for Jewish pride was that it worked, in ways that Hebraic laws didn't.
What does a fiercely tribal, immensely loyal, powerfully self-determined society do, when faced with evidence that a conquerer is doing a more effective job of running things, despite being of an inferior, heathen breed?
Revolt? The Romans were quite effective at putting down revolts. Despite greater social organization than in OT times, Jews were still no match for the highly cohesive legions. Rationalize the situation? Not an option, when the populace is illiterate, disorganized, and largely apolitical. It would be a monumental task to convince one's neighbours that somehow their subjugation was anything other than the failure of their almighty GD to protect them.
Luckily, there is another option: redefine what it means to be Jewish. 'Fulfill' the law, rather than obey it. GD cannot be the state; the state has failed, and GD doesn't fail. GD cannot be the Lawgiver anymore: Roman law had trumped Jewish law, and GD cannot be trumped. However, those issues are neatly and cleverly sidestepped, if one redefines the experience of GD as being more important than the rules. If we replace GD the triumphant King with GD the stern but loving Father, there is room to explain the socioeconomic situation. Commandments 1+2 become a stronger version of commandment 5 (or 4, depending on the particular religious interpretation) - 'honour thy mother and father, and especially THY FATHER,' if you will.
From this perspective, one can more easily understand the shift in method of adherence to law. The focus on the concept of an afterlife is a good example; in OT times, an afterlife just doesn't make sense. GD the conquering King was present on the battlefield; the smell of blood and glory was in His nose, in a very real, temporal sense. There was no time for metaphor and abstraction; GD was right there, sticking pointy bits of metal into other gods, even as a warrior representing him was sticking similar metal bits into representatives of other gods. GD could, and did, win - right here, right now, in a very tangible way. The OT view, unfortunately, doesn't work without the blood and glory of battle.
Enter 'heaven,' 'the Kingdom,' and other metaphorical abstractions, to save the day. You can't prove the strength of your lord by losing miserably on the battlefield: the trumpet of Gabriel only smashes down the walls of societies weaker than your own. But the Romans can't take away your love for an abstract representation of GD. They may ruin you, financially, judiciously, even physically - but they can't strip away your sure knowledge that your Father awaits you, and is with you always.
So, assuming that a)there is a monotheistic, Abrahamic GD; b)the Commandments are an accurate representation of His word; and c)that Word remained (and remains) valid, despite changing societal circumstances, and evolving concepts of religion (as divorced, slightly, from nationalism);
then
Human relations to GD have to change. When all you're left with to give to your Father is love, loyalty, and devotion, those become the natural sacrifices to offer - in place of blood, smoke, and glory. When you are unable to free yourself from the perceived yoke of oppression, you find a way to be a good slave, as true to yourself and your heritage as you can manage. So, without the currency of livestock surplus, military might, and literal/theological superiority to offer GD, you offer what you have. After all, obeying GD, and offering the best part of what you have to Him, is a defining characteristic of what you are, as a Jew, and Israelite, and as a human being.
tl;dr: there's no real difference between obeying the law because it's law, and obeying the law as a natural expression of devotion, loyalty, and love. As a toddler, we listen to dad when he tells us we can't have sour keys for supper - because he's DAD, and those are the RULES. As we grow, and are introduced to ideas and outlooks outside of the family unit we know, we still listen to DAD - not because those are the RULES, but rather because we respect and value the wisdom and knowledge DAD has, that we don't. But we still can't have sour keys for supper, either way.
Cheers.
