1) LA wouldn't tell you if your logic 0s and 1s are good. Last week I had a case of a bad logic 1s at 2.0V for a 4.2V device. Logic level translator went mad, and, obviously enough, a device couldn't make a good distinction.
Sure, a LA is not very well suited for solving analog problems, just like a scope is not the ideal tool for digital problems.
Oscilloscope is not well suited for digital.
LA is not suited for analog at all! Remember my example of a bad logic 1? It turned out that one of devices on a bus had too much of input current, and the PCB is still is a work-in-progress. If I did not have an oscilloscope, how am I supposed to discover that?
The longest record from LA I ever got was 90 seconds with 100 kHz discretization.
How do scopes in the same price bracket compare?
Scopes do this task even worse than LAs, but one do not expect a scope to perform well on this task. And I still do not have a slightest idea of why LA makers fail to put some more memory (and internal data compression) on their devices.
Never had any good experience with my cheap LAs. Unless you absolutely need to have 8-16 channels at once, it is wasted money.
It depends on what you call cheap. The ~$150 Saleae and Usbee devices work quite well, the main limitation being the channel count and 24 MS/s sampling rate. Decent LA software will have protocol decoders for most common serial protocols, you're not going to find these on your sub-$500 scopes. Note also that you can get a decent LA for much less money than a decent scope. Both are useful tools, and both have their place.
I do have a USBee AX clone. I was going to buy an original after trying it, but never did: it works as it is claimed to, but it's software is a fail in so many ways, that it still collects dust somewhere. As I reckon, my test device had some noise on the bus on power-up, and "USBee" triggered like hell, never showing me of what is going on after the noise came off. Putting a delay on startup was not an option.
Also, I have a chinese LA1016. It's software sucks even more, and it cost me about $450 a couple of years ago.
Surprisingly, the best one I have is an Open Bench Logic Sniffer from DP. It saved my ass a couple of times, but still if I did not have a scope - I wouldn't even bother to get LA.