In this particular case, the Seahawks had just lost TE Jimmy Graham for the season the week before, and as such needed to fill a spot for a TE. It seems like they signed one player, and then decided he wasn't a good fit, so moved on. They probably had to move quickly initially to get someone into practice right away, and so didn't have time to do a thorough job looking for a specific person - by the second week, they had more time to find the right person (or, two people, apparently).
This sort of thing is inevitable in the NFL, and happens frequently. Compared to Major League Baseball, replacing a player in the NFL is very complicated: there isn't a minor league to draw from, or a "system". There's a small practice squad, but that squad doesn't have very many players on it, so you often have to sign a free agent if you need to fill a spot.
However, NFL contracts are generally not guaranteed contracts - so there's very little cost to signing a player on a temporary basis. Teams will have a 53 man roster and 46 active players on gameday (7 inactive, which includes injured players who aren't sufficiently injured for the IR), so signing one guy and then cutting him to sign another guy doesn't really cost them anything: they'd still have that spot filled by someone, it's now just a different someone. I imagine there's a small cost in terms of paperwork and tax/etc. overhead (since NFL players make so much money, they hit the Social Security tax limit fairly quickly, even the lower paid players, so changing players every few games would have a higher tax bill) but that cost is insignificant compared to the hundreds of millions each team is making.
As far as why they want to make sure and have that full 53 man roster - even if a player is unlikely to contribute on offense, it's important to make sure you have a full set of players for special teams (And a 3rd string TE is very likely to be playing on special teams). While he wasn't activated, we don't know why that is; it may have been they just had him for insurance (in case someone got hurt), or it may have been that they wanted to use him but he didn't turn out to be talented enough or learn the system well enough.
Finally - don't discount the value of 'attending practice'. Having a full set of players for practice is important, so they can run plays - even if it means being on the 2nd string offense whose job it is to give the defense something to play against oftentimes. Being short a player means either a starter is doing a backup's job as well as his own, or they can't run plays as smoothly in practice. Plus of course the chance to see how good a player is.