Yes, the way that combat works is the same
I'll add a caveat. The PHB goes into more detail, and has had errata issued and posted at WoTC's web site. It's very much worth referring to as a number of details are addressed that were a bit "off" at original issue.
I'd recommend downloading them, and the Sage advice Compendium, for reference since these comprise official rulings form the game's lead developer: Jeremy Crawford.
As to comparing the combat rules to 3.5e: less complicated.
Experience: we began in the fall of 2014 where only the DM had the PHB and all of us had the Basic Rules to work with. Basic rules had enough info to run combat for three sessions before all of the party got their PHB, no problems. One of the things I liked is that the PHB and Basic Rules follow the same chapter organization: in both Chapter 9 is Combat, Chapter 10 is Spellcasting, Chapter 11 is the spell list ... and Chapters 1-8 all match up as well.
Significant Difference: feats
The Basic rules don't include feats. For that you need the PHB. The feats IMO add quite a bit of value to the game. The other big difference is "more spells in the PHB." The Basic rules have a smaller spell list than the full PHB.
I'd personally recommend the PHB: the latest edition should have all of the errata folded into it.
Update: The 2018 Basic Rules combined the Player Basic Rules and the DM basic rules into one pdf file, and includes a great many errata that have come up since the first publication.