Ascension is a form of shifting that can be performed by any strong person, regardless of species, though it is very hard to accomplish without working with Ki. A Limit Break, while not a shift, is still related.
What is Ascension?
Ascension is when you Ascend to a whole new level of power. A Limit Break is a temporary boost which runs out but Ascension is something that will be with you until you power down.
The best way I heard it described was on the old Ki Warriors site: Imagine you are a container (a bucket) that normally holds 5 gallons of water (this will be your Ki). When you use Ki your water level goes down, and when you power up it goes back up but never over 5 gallons since that's all you can hold. A Limit Break is something that allows you to hold 6, 7, or even 8 gallons of water (depending on how strong you are). However you can't get those bonus gallons back by recharging so once you use them you're back down to a maximum of 5 gallons.
Ascension is like getting a whole extra bucket so now your maximum is 10 gallons and you can keep recharging to that until you power down from your Ascended state.
What does Ascension feel like?
It's hard to describe. You feel powerful but not like with other techs. You don't just feel physically and energetically powerful, you also feel confident and in control and you know you will win.
What is a Limit Break?
Obviously, it is when you break through your supposed limits. These are often seen on those shows with "incredible videos." Tiny woman lifts a car, firefighter keeps going into the building to save someone even though there's not enough oxygen for him to physically keep breathing, etc. A Limit Break is when emotion and determination trigger extreme focus and strength (the Limit Break itself) allowing you to go beyond your limits, though only for a short while.
What does a Limit Break feel like?
There's not much "feeling" involved. You are incredibly focused on the task at hand. Looking back you realize you were moving faster and your attacks were stronger but right there during the Limit Break none of that registered. All that matters is winning.