Ok. Right. I'm going to do my best to not get too excited. Old green stuff is my bag.

Breathe.

On its use, and just to quickly amend your question (sorry!). Later Romans didn't actually use silphium - Pliny the Elder tells us in the first century CE that it is extinct, and only a single stem had been found in previous generations (Natural History 19.39). Inferior alternatives did exist, but they were not the original plant, nor as effective, according to Pliny. The majority of our evidence, though, is Greek, and the sheer extent of the evidence and our sources for it, which are not mythographers and are more gritty earthbound sources, make it clear that the plant did exist in some form.

Here's what Jacques André has to say on the subject in Les Noms des Plantes dans la Rome Antique (1985):

On the problem of identification, see the good focus of A. C. Andrews, Isis 33 p.232. Since Cato, usual.
Ill-determined species of Ferula from Cyrenaica for which Ferula tingitana L. has been proposed, and its juice, which served as a remedy and condiment (cf. Pliny 22, 100 etc and Diosc. gr. 3, 80). After the disappearance of the plant, we imported under the same name or under that of 'laser' the resin of an oriental ferule (Ferula asafoetida L.)

So, the two plants he identifies as potentially the sylphium is the Ferula tingitana, or Tangier Fennel and the Ferula asafoetida. The lean towards this as an identity is helped by coins which depict it, as in this instance. The identity of the plant remains "one of the biggest puzzles of ancient botany" though, according to Hardy and Totelin (2016, Ancient Botany). Hardy and Totelin identify seven different identities, including the Ferula tingitana mentioned above, and suggest follow up readings. So let's follow up!

First, they point us to a 2004 article from Amigues, available here. It is in French, and it is very long, and my break can only stretch so far. That being said, when I read it a few years ago, I remember thinking it was very good, so I'm mentioning it now. The second follow up is, mercifully, in English, and I can read English much quicker! Totelin's 2014 article offers a deep dive into the identity of silphium, especially p.33, which has a table of the various identities. This is the same list that (Hardy and) Totelin provide in Ancient Botany, but is much nicer to look at than the endnote of the book!

Totelin does direct us to studies done on the contraceptive capabilities of silphium, and we'll take a look at Riddle's 1997 book, Eve's Herbs: A History of Contraception and Abortion in the West, since it is a) open access (though the other two cited (Riddle 1991 and 1992) are available here and here), and b) simply put. Riddle assesses the effectiveness of an extinct plant by the effectiveness of the non-extinct alternatives, and we find ourselves coming back to the Ferula asafoetida and the Ferula orientalis. When tested with rats, the juice of these plants, which was used as an alternative, was found to be 40-50% effective at preventing conception. So, we can assume that the original product was more effective, though we do not know how effective. And, unless someone uncovers the exact plant, we are unlikely to ever know for sure. But it definitely existed, it probably worked as a contraceptive, and it was very popular with a certain randy Roman, who would keep on 'kissing' his Lesbia so long as they had contraception to hand:

You ask me, how many kisses of yours,
Lesbia, are enough for me and more.
As great the number as Libyan sands
Lie among Cyrene, the Silphian producing lands

Catullus 7.1-4

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Edited for formatting shenanigans.