Some sniper rifles seem to be upgrades of another. For instance I have a level 10 mantis and level 1 Widow. They both have the same reload time and close damage, when does the Widow surpass the Mantis? The Black Widow also seems to be the next progression of the Valiant; at what level does the Black Widow surpass the Valiant? Are there other rifles that are similar to each other like the Viper and the Indra?
You can find statistics for most snipers here: http://masseffect.wikia.com/wiki/Sniper_Rifles#Mass_Effect_3
The damage of Mantis and Widow is clearly different. Wiki data is often more reliable than the in-game charts.
The Mantis' advantages over the Widow are lower weight, and greater ammo capacity (10 shots total by default instead of 8). This stays true throughout the upgrade process.
The weapon upgrades have relatively minor effects, and the Mantis will never surpass the Widow for pure per-shot damage (Mantis X does 769.5 damage while the Widow I does 867). If you don't care about the extra 31 weight (nearly 50%), or the additional 2 shots you'll get with a Mantis, then you should definitely be using the Widow.
(Bear in mind though that extra damage may be overkill - if you're killing enemies in one shot with the Mantis, then the Widow has zero advantage. Depending on your targets and whether you have the thermal clips upgrade, being able to fire ten times instead of 8 is likely to be a big advantage in the first part of the game.)
I personally go with the widow all the way because of its pure strength and you don't have to add a special armor penetrating mod to it. It already has that capability naturally so you don't have to waste a mod spot. I started out with Widow 1 when I switched from the Matis and liked the Widow so much better. I'm now at Widow 10 and its a beast. Totally worth using. Just apply power amplifiers and Structural Ergonomics to your character and spec them with quicker recharge times. You may not even notice the slower recharge time.