Both cars that you have mentioned have their sets of pros & cons if pitted against each other. I'll highlight the good bits of the two. Might help you to zero down as to what exactly your looking for.
Pros for Hyundai i20
1. Much better looking interiors.
2. Much better overall external looks.
3. Exhaustive list of features. They indeed come handy while driving around.
4. Asta variant in your budget will come with the necessary safety features such as ABS
& Dual Airbags.
5. Overall quality of materials used inside the cabin is much better.
6. NVH levels are extremely low for a diesel engine.
Pros for Honda Amaze -
1. A bigger, more powerful engine.
2. Linear power delivery, minimal turbo lag.
3. Boot space surely bigger than i20.
4. Perception of a bigger car although the length of the two cars is almost similar.
5. Better fuel efficiency by the i-Dtech diesel engine from Honda.
6. Maintenance and service will be a tad cheaper as compared to the Hyundai.
Both cars seem fairly reliable and have been doing pretty good in our market, so you won't go wrong if you chose any of the two. Just go for the one which ticks more number of boxes as per your set of requirements.