Does the distinction between biblical versus rabbinic transgressions have implications or consequences today?
In the past, the punishment for them was different I believe.
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The most basic - and presumably most practically important - distinction between laws of Scriptural and Rabbinic origin is what to do in case of a doubt.
The general principle is: S'feika D'Oraisa l'Chumra, S'feika D'Rabbanan l'Kula - when confronted with a doubt in a law of Scriptural origin, take the stricter position; on a doubt in Rabbinic law, take the more lenient.
This impacts halachic decision-making on a daily basis, right down to the most mundane concerns of (for example) whether or not to eat a food cooked in a pot whose kosher status is in question, to make a blessing when you've forgotten whether you already have.
It is not necessarily the case that we'd say that a Scriptural transgression is "worse" than a Rabbinic transgression, if only for the fact that it's Torah itself that enjoins us to follow the teachings of the Sages - so that it's a Scriptural injunction to observe Rabbinic law, putting the two elements of Torah law on equal footing in the final analysis.
Even more so, however, the Talmud says, Chamur Divrei Sofrim mei'Divrei Torah - the words of the Sages should be observed (even) more strictly than the words of the Torah itself, for it is Rabbinic law that actually serves to protect the Scriptural; securing the Mitzvos from even accidental breach by setting the fence further away.
You're correct about the punishment. If someone carried in a "public thoroughfare" on shabbos (a biblical violation) because they thought it was Tuesday, they had to bring a sacrifice for atonement. If someone carried in a deserted forest (a rabbinic violation) thinking it was Tuesday, no sacrifice would be brought.
If done intentionally, warned, and witnessed, (plus a bunch of other caveats), biblical prohibitions could carry all sorts of punishments. Rabbinic ones generally (barring extreme circumstances) would be "whip him until he says he's sorry." That's how things were a long time ago.
Today, we generally say that a biblical violation is worse than a rabbinic one; and that mistakenly violating a biblical violation requires repentance (regret, confession to God, resolve to do better), whereas mistakenly violating a rabbinic one doesn't (but please educate yourself better, or buy a big calendar or whatever it takes to prevent the problem from happening again).
There are many other distinctions in the law between rabbinic and biblical prohibitions as they pertain to "may I do X", not "what if I did X"; for instance there are many (not all, it's complicated, please consult a rabbi) rabbinic prohibitions that had built-in exceptions for non-life-threatening medical or military situations. (If it's life-threatening, don't call your rabbi. Do like Tuesday and call 911 right now. Or go shoot the bad guys, or whatever.) For instance: "It's Shabbos and I have a really awful migraine and I'm in bed the whole day because of it." May I light a fire to make myself more comfortable? Biblical prohibition, no. May I take an aspirin? Rabbinic prohibition, they left an exemption for the seriously-in-pain, so yes.
if there is no difference in speed or quality of care, that one should choose to do a rabbinically prohibited action, instead of one that is Biblically prohibited.
(Note the "if there is no difference". Otherwise, saving a life takes priority.)
Note that, as Daniel pointed out in a comment on this answer:
I would imagine that it's probably the case that if for some reason it becomes impossible to avoid doing a prohibited action, a rabbinic prohibition is preferable to a biblical prohibition even not on Shabbat and possibly even not to safe a life.
But I have no source for that at the moment.
Besides for all the other good answers, there's another issue, mentioned by the Ramban in וָאֶתְחַנַּן - Devarim 4:2 on the Passuk forbidding one to add or subtract Mitzvoth - לֹא תֹסִפוּ עַל הַדָּבָר אֲשֶׁר אָנֹכִי מְצַוֶּה אֶתְכֶם וְלֹא תִגְרְעוּ מִמֶּנּוּ
When discussing how to deal with Rabbinical additions, the Ramban states:
ומה שתקנו חכמים משום גדר כגון שניות לעריות וכיוצא בהן, זו היא מצוה מן התורה, ובלבד שידע שהם משום הגדר הזה, ואינן מפי הקב"ה בתורה
One should be aware what are Rabbinical additions to the Mitzvot.
Although they are [real] Mitzvot (since it's a Mitzva to listen to the Rabbinic decrees), nevertheless one should know they are not Divinely decreed, but added as a fence.