There are some relevant points to consider when seriously discussing routine access to Mars
1) Pace of technologic development and cost: Gagarin to Armstrong/Aldrin was 8 years, but the Apollo craft was very delicate, very expensive, pushed every envelope to the breaking point and was therefore very dangerous despite the relatively few serious accidents given the nature of the beast.
2) Moving from experimental to production technology: One goal of the new companies is to make the craft more reliable, use less experimental technology, and make it more comfortable. Both Apollo and Shuttle are experimental technology that require large staffs of highly trained people to operate; the ground operations for Shuttle are a significant part of the program. Early autos required frequent maintenance (tire repair, water, oil, etc...) and still broke down frequently. Today cars go forever despite the lack of maintenance we give them. Form factor aside the Orion and Dragon craft are as completely different than the Apollo craft as a Model T is from a Honda Civic.
3) Sustainable interplanetary infrastructure: for reliable access to Mars we need to invest in something like the Aldrin Cycler. A quick shot to the Mars, like the one to the Moon, is akin to the Lewis and Clark expedition. To open up the frontier the government, decades later, invested in the railroads which provided frequent, reliable, and sustainable round trip capacity. They took longer to build than the carts and boats that took Lewis and Clark, but made settling the West possible.
Just my 2c