There really is a difference. This category of piers has a dozen names, each with a very narrow, specific design, but also used in the broader sense for general foundation support and repair.
Let’s be a bit narrower and define the differences and applications.
A helical pier is a square bar or a pipe that has helices (looks a lot like a post hole digger). A hydraulic motor turns the pipe causing it to screw into the ground. The direction could be vertically or at an angle (degree of batter) as required by the design. The stiffer the soil, the greater the load bearing. It has the unique characteristic of having the same capacity in compression (vertical load) as in tension (tieback in a wall).
It can be installed to capacity with the installing equipment, exclusive of any building, foundation or other mass.
It is ideal for new construction, light loads, fractured footings, tiebacks, tension loading, temporary applications (easily removed), and has generally maximum real capacity of 156 Kips, (1 kip = 1,000 pounds) although there are a few sources with larger capacities.
What about push piers, (also called resistance piers)? These are open-ended pipes that are pushed vertically into the ground until these terminate in a soil determined to be strong enough to hold the load these are intended to hold. A hydraulic ram is used to insert the piers then anchored to the structure. The weight of the structure becomes a resistance mass to push against, thus the name, push piers. Capacity is limited to the direct load the building provides. It is a one to one capacity with no factor of safety (FS). FS comes with multiple push piers that, when designed properly, share the load. Full capacity can only be reached if the reaction load is heavy enough to resist it
A push pier is a strong pier, though very simple. It is stable and ideal for heavy loads. It can be installed in a very small area, and can underpin an existing foundation that might be undercut by an adjacent excavation. Typical maximum capacity is 90 Kips.
There are overlaps in some application such as nominal to fairly heavy remediation projects and underpinning. A push pier or a tieback cannot be removed because their sections are compression fit only. Neither can it be a new construction pier because it needs the reaction mass in place to push against. It should not be used in light load (because of the minimal weight) or with cracked or broken footings.
Helical piers can do most any remediation work because it creates its own capacity exclusive of the weight of the structure and does not “load-release-load-release-load” the footing like push piers. It’s not as good as a push pier for underpinning because it cannot be sleeved to decrease buckling. Helical piers are more expensive but go in faster. Most of the time, they use the same footing brackets.
Both are vibration free, cost effective, long lasting, engineer and building inspector approved, and have 30 years and more of design, field testing, and actual field use. Properly installed, failure rate is so small as to say virtually non-existent. Most failures are related to other factors. All piers can be easily re-loaded for proper performance.
Whether your installer chooses a helical or push pier, rest assured that it will be a gratifying decision and a great solution to your problem.