I've own a Bike Friday Pocket Llama and a Dahon Speed D7 at the moment.
I have also owned a Dahon Mu P8 in the past, and I have extensive experience with the Bike Friday Tikit.
The aforementioned comparisons are pretty accurate, but there are a lot of differences still.
Simplifying Bike Friday's line, they really make 2 styles of folder: "Suitcase folders" (their pocket bikes),
And their commuter folder (the Tikit)
It sounds like you experienced the Tikit. As much as I love my Pocket Llama, it is neither "slick" nor "really compact" while folded, rather, its a big jumble of metal and loose bits
The Mu P24 and Tikit are very different bikes, however. They aren't really even designed to fill the same niche, if you are comparing stock models. I'll try and compare some of the more important things.
The Mu P24 is best suited for someone who wants a folding bicycle, but lives in a very hilly area. It has a much larger gear range than a stock Tikit with a lonely 8sp cassette. The Mu P8 is a much more even comparison in this regard.
The Mu is equipped with 20" wheels, whereas the Tikit uses 16" wheels. I personally still enjoy 16" wheels, and tires are readily available - but a few more choices are present in the 20" class.
The Mu P24 requires you to lower the seatpost, move the stem riser around / twist the bars in order to fold the bike. The fold is pretty manageable though, and the bike stays together while folded. You can sometimes roll the bike on its edge, if the handlebars aren't poking into a wheel.. Its generally not a rollable bike, I find.
The Tikit does not require seatposts moved or anything of the sort, all of the dimensions are saved. No re-adjusting after folding. The Tikit folds exceptionally quick, especially the hyperfold model, and the bike is rollable while folded with a handle on the back, making it very easy to deal with. There is even a one sided rack available that will hold a pannier and let you roll it around while folded. To me, this is a pretty great advantage.
Warranties and customer service wise, I find Dahon to be pretty decent. There's a 5yr / lifetime warranty from Dahon. I have a Speed D7 that was affected by the recent folding stem recall, and it wasn't a hassle to get the service done.
Bike Friday though, is like nothing else. You can call in and talk to specialists who can help you out with anything you want. They'll upgrade the bike, customize it, and provide you complete support as long as you own the bike. I've found their service to be exemplary.
And of course, the Tikit is customizable. Dual Drive is also available, as are many upgrades.
Edit: Oh yes, the 2010 Tikit has been redesigned to be a bit stiffer too.