Can I install more apps if I get more internal memory/storage? Most modern top-level Android devices, e.g., Samsung S4, has:


2GB internal RAM
16/32/64GB internal memory/storage
up to 64GB exernal/removable memory/storage


Does the amount of internal memory/storage make any difference - for instance how many apps you can install or does the size of the storage only affect the number of images/music pieces/... you may download to the device?

I agree with the answers on the performance part but what the number of apps?
The answer is...It depends.

As a rough equivalent between Android and Windows PCs:


RAM = RAM
Internal Storage = C: drive
External Storage = D: drive etc...


Allocation of RAM works differently in Android, vs. say Windows or Unix. The usual notion of more is better doesn't necessarily correlate here. 

As I understand it, it's diminishing returns once you go beyond what's required. Android basically tries to keep the apps/files you use most cached in RAM. Swapping in and out as it finds necessary. Once you're beyond what's needed, then it seems like Android just starts caching random and useless things. That's why if you use a Task Manager on your phone there always seems to be "stuff" running that you might feel inclined to just kill and free ram...But there's a lot of debate that says you should leave it alone and let the Android OS handle it.

As for Internal Storage, I personally go for the biggest I can get in my budget. The Android OS files go there, all your settings, music, photos, and apps by default are installed here. Nowadays though internal storage offerings from manufacturers are larger and larger and options to put files on external storage are getting better and better.

Considering today's offerings and requirements, I'd personally suggest when buying a new phone to get:


At least 1GB, 2GB preferred (As new apps with more bells and whistles come out, their RAM footprint also increases. 2GB is still probably way more than most average people need, but it's quickly becoming the standard on newer phones anyway.)
Internal Storage at least 16gb (Again, as new apps come out with more bells and whistles and cameras with higher megapixels and recording High Definition movies, they all keep demanding more space. The more you have now, the less you'll have to deal with later if you have to start cleaning up or moving things off to external)
Get whatever the cheapest option offered with the phone (if it's offered) (Buying the external storage from a 3rd party is much less expensive and likely faster and more reliable).