Which one allows me to do more customization? I know that there are lots of templates for Perstashop and thats a plus.
What are other main differences?
For me, I choose bigcommerce.
My reason is very simple, because I’m not good in programming.
With hosted solution like bigcommerce, I only need to plan one of their plan and everything is there.
No need to do any programming or tweaking. 
Well it depends on what kind of customization you are looking for. If you are using auto build ecommerce platforms then probably, you would not get a chance to do more customization. Some ecommerce platform do not allow any customization.
PrestaShop and BigCommerce are similar to that.
If we consider Magento then probably it will allow you to use default customization which is not their with PrestaShop and BigCommerce. So just elaborate about the customization and you will get you answers.
I’ve never used either, but both seem to be fully customisable, according to their sites.
I have worked with the company where this issue was raised. Client’s website was in PrestaShop and he wanted to migrate it to Magento just because he wanted some customization which wasn’t there with the PrestaShop.
It is really depends on your needs, resources and skill.
If you are good in programming and able to setup the server (VPS or Dedicated), it is fine to go for PrestaShop. Because it is open source, so everything can be customized (if not default function, you can still coding it by yourself).
Otherwise, if you only want to sell your products and focus on growing your business. You should avoid or eliminate the tasks (e.g. server setup, monitoring, maintenance, backup plan, bug fixing…) that cannot rise your sales and go for hosted service.
It is good to use Bigcommerce if you have lot of products.
But if you just start to test the water, you may also consider some other hosted online store creators for small business starters.
shopify
jooy
bigcartel
Some of them are providing free/trail plan.
The main difference is that Big commerce is hosted and you can only modify it so much. Prestashop is totally customizable so you can do anything you can imagine with it.
Your development team just did not know how to work with Prestashop. You are placing a limitation on the software that is actually a limitation of your development team. If you want to elaborate on the issue, I will be glad to tell you how it could have been worked out.
Hi, I am Community Manager for PrestaShop, I can help answer any questions you may have about our software. PrestaShop is Open-Source, that means the Code is available for all to see and use as they wish, we have a GitHub dedicated page so developers can contribute directly to the software, https://github.com/PrestaShop/PrestaShop
First, PrestaShop is a freeware, used by over 150,000 online stores. you can use the entire software without ever paying us a cent to use it. Yes, we have additional Addons and templates that we sell, but the out-of-the-box feature set, rivals,if not exceeds other paid software. We are not a hosted solution even though most quality hosting providers have simple one click install. Do you have to know programming to use PrestaShop? Absolutely not, however, due to the open-source flexible nature of the software, many people chose to have Developers to further customize their website. We have offices in both Europe and the United States and really try to help out our merchants as best we can. I would suggest that you check out our new v1.6 that will be released early next year, http://vimeo.com/75405291 , as well as our highly active forum at www.prestashop.com/forum
Please let me know if you have any questions, I’ll be happy to help.
Cheers,