Please can anyone explain me, what TestFlight i should use to test my app. First - TestFlight from apple and second - TestFlight from http://www.testflightapp.com/
You can use either. They provide different features based on your needs.
TestFlight from Apple - There's actually two flavors: internal and external.
Internal Distribution: You can distribute your app immediately to anyone registered as an iTunes Connect user for your organization with the privilege levels of Admin, Technical, or Legal. Now those are elevated permissions, so that won't make sense in a lot of cases. The benefit of internal distribution is that your app can be sent to the users immediately. The number of users for internal distribution is capped at 25 as well.
External Distribution: You can send your app to as many as 1000 external users. They don't need to be iTunes Connect users for your organization either. The drawback is that Apple requires review of your app, although I've heard that beta review happens a lot faster than formal app review for app store distribution.
With either internal or external distribution, your limited by the number of users, not devices. This is a major advantage if your uses have multiple devices (like someone wants to test on both an iPad and an iPhone - they only count as one user).
TestFlightApp.com - Yes, Apple acquired them. It looks like they are still operating as a separate service. It's unclear how long they will be around, and that's a risk if you are looking for a service to use for the long term. The benefit of testflightapp.com is that you'll get a lot more functionality from their service than Apple provides (stats, better control of distribution, crash reporting, etc), the main drawback though is that each devices you want to distribute on requires you add the device to the iOS Developer Portal, and you're capped at 100 (for all your apps), and it's only reset annually.
Two other good alternatives to testflightapp.com are Hockey App, and Crashlytics Beta.