I've used both. Both are pretty good.
Linklicious is good in that you can use unlimited API calls/integrations on their higher priced package. This is good if you use a lot of automated SEO services. If you just use one of these types of services, their basic plan is the way to go.
Lindexed also offers a similar service, but with strictly limited 1 API call (same as Linklicious basic plan). You cannot choose a higher plan for unlimited calls, which means you need a new Lindexed account for each SEO service you use that you want to draw/call backlink reports from. With Linklicious, if you get the pro account - you can call unlimited amounts.
All in all, they both seem to do their job properly.
My only complaints:
Lindexed: I assumed there were unlimited API calls on their $20/m plan as they made NO mention of how many API calls are able to be made. So I assumed I could use all of my automated SEO service accounts and get all those links called in via Lindexed. Turns out they have a limit of only 1 account to be called in - kind of annoying, as I assumed it was unlimited because they made absolutely no mention to any such restriction on their site.
If you use just one such service, it's fine - but I use more than one (or have more than one account with an SEO service) so it would be very useful for me.
Linklicious:
Signing up a new account through them seems tedious. Not sure if I recall correctly, but it was more involved than the usual way of just make an account (username, password, email etc). It required me to do more, so it was actually the reason I went and chose Lindexed because, as silly as it seems, I was pissed off that they had not made it easier to just go in, make a new account and get sh*t rolling.
In hindsight, I should have stuck with Linklicious because it looks like I'll be needing them soon anyway, with their unlimited API calls.