There's no quality difference to where the recording is stored - if you plug the same mic into different recording devices, it should record the same. Different quality mics (and recording environment) will give you more difference to your sound than anything else. Do make sure you are recording uncompressed (WAV) format though, using compressed formats will obviously reduce your resulting audio quality.
I have some external recorders, I use them mainly because you can plug in a large SD card, hit record, and the device will just sit there and do that job for hours, no issue. I don't have to worry about a piece of software or laptop crashing, they're portable and I can plug them in at the back of a sound desk or anywhere I like, etc. If you're recording lots of audio, they're just handy to have and use. Most external recorders do default to higher quality settings than what Audacity might (i.e. mine default to 24-bit/96kHz), but you can change those settings in Audacity as required. Portable recorders can also have other interfaces and features that come in handy if you're using them in lots of different environments - I almost always just use mine with the same standard setup, and most of the extras go unused!
For what you're doing, hooking the mic into a laptop and using Audacity will do the same job just fine. Just make sure to monitor things and check your recordings - make sure audio isn't being dropped due to inability of the system to keep up to speed, and that everything is recording as you expect (mic cable/connection good, levels good, mic close enough to the source to pick up the sound cleanly, etc).
You might also consider adaptors for your phone:
I've not personally used these, but there are various adaptors out there now to interface sound inputs direct to your phone, as well as dedicated external mics (the above adaptor gives you a more generic approach that you could plug anything into, but there are specific mic only devices for phones as well). I would just double check how the quality options would vary on a phone recording vs using Audacity dedicated on a laptop, which might not even matter for the types of recordings you are describing.