They are QUITE different
They serve both a different purpose, AND contain different content.
Newbie Set vs. Legal Document
The Basic Rules were released prior to the SRD, and are designed to provide a very limited set of gameplay material in order to give players a taste of D&D 5E before they purchased it.
The SRD is a legal document intended to provide all of the 5E rules and material that Wizards of the Coast consider to be Open Source material that is legal to use under the Open Game License (OGL). You may read the details of this license at the top of the actual SRD pdf
In other words, if you want to create and sell material for D&D 5E, then you may only use material that exists in the SRD or material that you invent.
For example...the iconic D&D monster The Beholder does not exist in the SRD. Therefore, if you want to create content under the OGL, you may not use a Beholder in that content.
Differences in Content
The SRD and the Basic Rules contain dramatically different amounts of Content.
The Basic Rules only includes content for 4 classes: Cleric, Fighter, Rogue and Wizard, each with only a single sub-class (Life Domain, Champion, Thief, and Evoker). Only 4 races are offered: Dwarves, Elves, Humans, and Halflings.
The SRD, on the other hand, contains every Race and Class from the PHB, with at least one sub-class per Class. Additionally, there are several Monsters included in the SRD that are not in the Basic Rules, such as the 'Tribal Warrior.'
Going the other direction, the SRD includes precisely one Background: The Acolyte. The Basic Rules contains 5 backgrounds.
I suppose the best way to show the difference is to simply share what Wizard of the Coast says about them...
Regarding the Basic Rules...
The Basic Rules is a free PDF. Anyone can download it from our website. We want to put D&D in as many hands as possible, and a free, digital file is the best way to do that.
And the SRD
The Systems Reference Document (SRD) contains guidelines for publishing content under the Open-Gaming License (OGL).