You can apply for both. They won't pay you both at the same time, but if you meet the criteria for both payments, they will offer you the more advantageous. However, more likely they will refuse both applications and you will have to appeal. However, the appeals process has a high success rate and you may be awarded one or the other after appeal.
Invalidity Pension requires at least 5 years work in Ireland over your life to be paid the full rate. Disability Allowance is means tested. If you have a partner/spouse, their income is taken into consideration.
Disability Allowance is easier to get from a medical point of view. You only have to show that you are substantially restricted from work which you would be suitable for you, in light of your age, qualifications and experience. You only have to show that the restriction will last for 12 months. You have already been out of work for nearly at least 1 (and probably 2) years if your Illness Benefit has nearly ended.
For Invalidity Pension, you have to show that you are incapable of all categories of work (including light work, sedentary work, or work that you have no qualifications or experience in). You have to show that you were incapable of work for at least 12 months before you submit your application, and likely remain so for 12 months after the date of your application.
Invalidity Pension is better financially in most circumstances, but there are a small number of circumstances when it is not, eg. if you have a spouse/partner who is on Jobseekers Allowance, or if you are relying on contributions from other EU countries to qualify.
Also should say that applications for Disability/Invalidity take about 3 months to process. The appeals process takes between 4-6 months in most cases. If you have no income while you wait for the claim/appeal, you can apply for Supplementary Welfare Allowance, or if you have a spouse/partner is on a separate social welfare claim, they may be able to claim for you on their claim while you wait.
You should also ensure that you continue to submit medical certificates, even if they stop payment. This ensures that you continue to get credited PRSI contributions and also means that the Department continue to accept that you are incapable of work, which will assist greatly at any appeal.