But I am confused about the differentiation between sms and messaging apps?
A "messaging app" is a general purpose designation for any app that can send a message from one computer to another (or to several). It doesn't specify what or how.
SMS is a particular protocol (see
https://www.ietf.org/rfc/rfc5724.txt ) and the specifications in that document have to be met for the message to be SMS. (If it's not, some SMS apps will ignore it, some will display it incorrectly, etc.)
iMessage, the system Apple uses, is another "messaging service", but it's not SMS. (It
can send pretty standard SMS messages
if you tell it to send SMS to a particular number at least once [at least, as of November 2016] - but the default protocol is a proprietary one and non-Apple phones either get nothing or they get something that looks nothing like a message. (And the recent "my messages are all coming out in Chinese text" complaints could be part of that - but without being able to see the specs of the protocol, who knows?)
Is it possible for someone(s) to explain why for instance more people do not use the Verizon app?
Proprietary apps - Verizon, iMessage, etc. - mean that both the sender and the receiver must be using that app as their messaging app. SMS means that anyone can use any SMS app. (It's like 2014, when sending an attachment on an SMS was only possible if the recipient was on the same carrier as the sender.) No one wants to keep 10 different messaging apps on a phone and keep switching between them as to which is the default receiving app, depending on who's sending you a message (if you could even keep track of that - or if all of them could also receive SMS, which they can't).
Is there something I am missing regarding text messages(is that the same as an sms?).
Since it's not, yes. You're missing the fragmentation that's occurred since everyone and his brother wrote a different "text" protocol.
All my friends that have iphones or android phones seem to get my messages quickly.
IF an iPhone sends an SMS to an Android phone, it'll be received. If it sends an iMessage (the default in iPhones), the Android phone gets nothing. As for speed, that depends on the server involved. Verizon's servers are probably pretty fast - but "Joe, down the block"'s server? With a 300kbps upload speed trying to handle 300 messages at once?
The Verizon messages seems to work over wifi as well as the mobile connection if necessary.
That's Verizon's proprietary stuff, it may or may not work if you put a different SIM in your phone, or put their app in a different phone (one on a different carrier). In fact, it probably won't work - unless Verizon app-to-Verizon app-protocol is standard TCP/IP protocol - in which case a standard SMS app won't talk to it.
I came to this question because I was looking at Allo but people were stating that it does not do sms
No, it does Allo, a different protocol. It still sends and receives text messages (and other forms of communication), but both people - sender and receiver - have to be using Allo as their default text app.
and I realized that I do not understand what sms is compared to what Allo does.
Allow doesn't use the specifications in RFC 5274, referenced above - which
is what SMS is. (Almost all protocols are spelled out in one or another RFC document - and usually referred to by the document's name by people in the industry - which is why "3 wire, 5 volt, RS-232C" seems so funny to the industry - the RFC
specifies at least 9 wires and 12 volts, so ""3 wire, 5 volt, RS-232C"" and "round black top on the square white bottom" are about the same thing - something that can't exist. SMS is one particular format for a message.) Aside from the addresses and messages, all SMS messages look the same. Allow looks different. iMessage looks different than either of them. So does Hangouts.
You can come up with a dozen different protocols to do the same thing - but they're not "interoperable" - one can't receive, or usually even detect, messages from another.
Then I thought, wait, perhaps Verizon messages is not sms also?
The fact that it's not sent over 2G is enough to make that a certainty.
And if not, does it matter?
It does - unless there's a fallback to SMS in the app. But if there is none, if you send a Verizon text message to me I'll never see it, since I don't run the app. I do have a few SMS apps I can use - but only SMS apps, I don't play with every text app that pops up for a few months.