Sooo, Lute and Gerik. Lute is a badass magic-using prodigy. Gerik is a mercenary.
Enough said.
So anyway, Lute joins in Chapter 4. Gerik joins in Chapter 10. This gives Lute an six chapters of availability over Gerik, which is a huge bonus. This is almost a third of the game, giving Lute a ton of opportunities to get Experience that Gerik simply doesn't get. In addition, he extra availability is during the earlygame (the hardest part of the game), making her more useful during the early game than Gerik is during the midgame when he first joins.
Let's look at how Lute performs when she first joins.
17 HP
6 MAG
6 SKL
7 SPD
8 LCK
3 DEF
5 RES
With a Fire tome, that gives her:
11 ATK
6 AS
22 AVO
3 CRT
110 HIT
8 CEV
Those are plenty good stats for the early game, she'll almost definitely make it into your party for the next several chapters. Let's look at Lute's performance in the earlygame:
Chapter 4: Lute probably won't get to do much in this chapter as she joins in the middle. She's plenty useful once she joins though, as she's the only character you have besides Artur (who's probably busy on the left side of the map) that can hit the monsters on their weak Res, easily 2RKOing or 1RKOing with a bit of Neimi or Eirika support. Let's say she gains one level from picking off weakened enemies and weakening enemies.
Chapter 5: Well, Lute's moderately helpful here I guess. She can easily pick off weakened units once again, but she's not especially helpful. She can help beat the bandit reinforcements if two units block the passage, or go up the left side. I think she can gain another level here.
Chapter 6: Unfortunately, in the early stages of this chapter Lute's more of a liability than a help, as her weak Defense means she gets 1- or 2RKOd by nearly everything in the chapter. However, she can be quite helpful once your party starts advancing and you have a good defensive line, as hitting Res means that she'll be doing more damage than a lot of your party. I think it's safe to assume she gains another level or two from this.
Chapter 7: Lute has MAJOR problems here. Ballistas will destroy her if they hit, cavaliers that can outrange and kill her run rampant, and even the boss has a Javelin. Unfortunately, I don't think she's getting any levels here.
Chapter 8: Now this is where Lute can really shine. The many chokepoints on the map provide ample opportunities for Lute to snipe from behind cover, especially as she can ORKO Knights after Seth or another unit softens them up a bit. She's probably one of your best bets for taking down the Mage reinforcements, and can even hit Tirado if you get him to equip his Silver Lance. She can easily gain two, maybe even three levels from this chapter.
At the route split, we take Eirika route.
Chapter 9: Lute is fairly helpful in this chapter. She can easily take on Pirates and Mercenaries ORKO most of them, but the wandering Snipers can and probably will ORKO her. She can probably get another level out of this.
So let's say we have a Level 9 Lute at Chapter 10, when Gerik joins at Level 10. I'm going to compare their stats here, Lute is the first column.
20.6/32 HP
11.2/14 STR/MAG
8.4/13 SKL
10.6/13 SPD
11.6/8 LUC
4.2/10 DEF
8.2/4 RES
We can see here that they're decently matched, with Gerik having a slight lead. However, although it may appear that Gerik has the damage lead here, Lute's hitting Res instead of Def, which makes a huge difference as most enemies at this point have much higher Def than Res. Skill is pretty much a moot point, as both Gerik and Lute will be hitting almost all the times because of Anima's high Hit rate. The Speed difference lets Gerik double a few faster enemies, but Lute will still be doubling the majority of enemies she fights. The Luck differential is largely pointless, and Lute's extra Res lets her combat Mages better but she'll get murdered by any physical hit. That being said, one of Lute's biggest advantages at this point is her reliable ranging: as many enemies you fight midgame are Cavaliers, Soldiers and other things with 1 range, magic is truly a godsend to help deal with them. In addition, a very large number of enemies in the midgame carry Lances, leaving Gerik at a huge disadvantage unless he promotes early. However, magic users are fairly uncommon (especially as most of them are Mages), meaning that Lute can help against them too with her huge Res. Gerik can generally ORKO Mages too, but his low Res means he'll take huge amounts of damage if he can't straight out OHKO.
Now for some more Chapter-by-chapter comparisons.
Chapter 10: Gerik starts out isolated and won't be doing ANYTHING until you can get Tana to his stronghold. Once you do though, he'll be pretty useful for clearing out the bottom section of the map. Lute can help you clear a path to the gate, but won't be too much help in the final assault as the Ballista destroy her and she can't really do anything to Pablo. Also, note that Pablo drops your second Guiding Ring, and as Gerik comes with a Hero Crest neither of them struggle for promotion. I would say this Chapter's a TIE overall.
Chapter 11: Unfortunately this chapter scares Lute. Granted, she can pick off monsters easily, as they have pathetic Res. However, archer Bonewalkers do heavy damage to her, and Venin weapons really mess with her day. I'm giving the win to GERIK here.
Chapter 12: Lute and Gerik can both be fairly useful in this chapter. While Gerik can help smash your way upwards, Lute can pick off dangerous Baels while they're still in the mountains, and damage basically anything on the map severely. In addition, by this point you'll have many physical tanks that can block paths easily, but Lute is basically the only thing you have for picking off Baels in the mountains. Although it's close, I would say LUTE wins this chapter.
Chapter 13: Welcome to the BRAWL. All shit aside, if you're going straight for Aias (I'm gonna assume you are), Lute's a bit more helpful than Gerik here. Reason being, if you can take out the Ballistas early (not hard if you use Seth or another Cavalier/Paladin), she can basically ORKO everything guarding Aias, which Gerik will sometimes fail to do as some of them wield Lances. As for the boss, Lute and Gerik can both damage him if he's not using his Short Spear, so pretty much even there. That being said, LUTE probably edges Gerik out just a bit here.
Chapter 14: UGH. Lute's very vulnerable in the first half of this chapter. What with the Snipers in the entrance hallway and the Shaman reinforcements, she'll have a hell of a time getting through to the top part of the hallway. Gerik has a significantly easier time of this, as the Shamans can hit him but the Snipers pose much less of a threat due to his large physical bulk. However, in the second half of the map their roles are reversed. On the approach to the throne room, Lute can easily bash through the Knights in the way that Gerik can't do anything to. Lute also fares slightly better against Caryle as she can fight him at 2 range, so she won't get murdered by his crits. Honestly, I'm calling this one a TIE.
Chapter 15: Good times are here again, Lute DOMINATES this map. She has full movement range, while Gerik can barely move. Lute comes in handy for fighting down the middle of the map, destroying the low-Res Fighters and Cavaliers, and generally killing everything (besides flying units) that need killing. LUTE easily takes this chapter.
As a side note, I'm assuming both Lute and Gerik are definitely promoted here, if not a chapter or two before.
Chapter 16: Both Lute and Gerik face some problems here, assuming they go with the main group. Gerik's main threat is from the two sages with Purge and Bolting in the throne room, both of which can do heavy damage to him. Lute faces a ORKO from any of the dangerous physical threats in the chapter, but you'll have more than enough tanks to allow Lute to stay behind a barricade and hit enemies' weak Res. Other than that there's not much to say about their performances. Just based on the fact that LUTE has less crippling flaws, I'm giving her the win here.
Chapter 17: A LOL chap in any case. Lute can easily take down most of the Sages with an accompanying healer, while Gerik can blast through Warriors with no trouble at all. Honestly, there's not a huge difference in usefulness here. 'm going with
Chapter 18: Gorgon eggs, wheee! In any case, honestly Gerik isn;t doing much this chapter to stand out. Sure, he can ORKO Gorgon Eggs and ORKO Gorgons safely, but then again so can everybody in your entire party (pretty much). Lute, with a tome like Shine, can do the same thing, but from 2 range and will barely take any damage in the process! I think this makes this chapter a pretty clear win for LUTE.
Chapter 19: This chapter...anyways, both Lute and Gerik haves roles to play here. Gerik can block chokepoints and take hits, while Lute can snipe enemies from behind, heal, and take on Riev better. I don't think either character has a distinct advantage over the other, so I'm going to say TIE.
Chapter 20: Here's where you want to start using your Sacred Twin weapons. Lute has practically NO competition for Excalibur, she's going to get it. Gerik, however, faces heavy competition for the Audhulma between Seth, Joshua, and your other Paladins, while many other units can make better use of the speed boost from Garm.
As for the chapter itself, both Lute and Gerik can assist in blasting through the swarms of enemies. Lute is far better for killing Cyclopses and other physical monsters, while Gerik takes on Arch Mongalls better. However, as Lute will probably have a Sacred Twin weapon and Gerik probably won't, I'm giving LUTE the win here.
Endgame: At this point, it comes down to whether Gerik has a Sacred Twin weapon or not. If he does, he can basically ORKO the enitre chapter (which Lute is doing with Excaliber anyway), meaning that they're roughly evenly matched; if not, he'll lose. However, LUTE has higher Res to help her deal with Draco Zombies, which I think gives her a narrow win over Gerik.
So there ya go; it's not great (or even good), but it's there.