In fact, 必要 is not a verb, but a noun. It can be an adjective if it is followed by な.
As a noun, 必要 means "a necessity" or "a need". As an adjective, it means "necessary" or "needed".
Is there a necessity to buy such an expensive computer?
たくさんの文献を読む必要がある。 There is a necessity to read a lot of literature.
ログインが必要です。 The login is a necessity.
But 要る / 要ります is a real verb, so the construction will be different:
パソコンが要ります。 I need a computer.
I would say that the meaning is quite similar, and that the main difference is their role in the sentence (verb, noun, adjective).
A slight difference about the meaning could be that 必要がある is more used in order to express an "external necessity" (that doesn't come from you), like a rule or a law. For instance, if you want to buy something, there is a necessity for you to have money. On the contrary, 要る can also express that you want something, so this is an "internal necessity" (that comes from you).
As @naruto said in the comments, the expression "を必要とする" ("require" / "call for") can also be found:
パソコンを必要とする。 This task requires a computer.
技術を必要とする分野 a field that requires skill
As @Chocolate♦ said in the comments, "読むことが要ります" would be understood but it sounds strange and is almost not used. Prefer other suggested alternatives.