Do I need to create a whole new form for each client when small details need to be changed?

I have a standard photography contract that I want to use with all my clients. There are small details that will need to be changed for each client, for example the session fee and what specifically is included in the package they chose.

Do I need to create a whole new form for each client? 

No you won't have to. Though the workflow would highly depend on what exactly you wanted to do. Since we have no idea on what you're trying to achieve, I'll just list 2 scenarios and cover them both:

1. If you will be asking them for a payment straight from the form, then you'd need to use a Payment Processor.

Here's a complete list of all payment processors we integrate with: https://apps.jotform.com/category/payment_processing

In case you're not sure which one will work for you, I recommend reading this article: Which-Payment-Processor-is-Right-for-Your-Business 

Once you know which one you'd like to use, just setup your products as you please. All payment processors have an option to add multiple products (in your case, sessions fees) with different prices, options, etc. I strongly urge you go over our compendium of guides about Payment Forms to learn more.

2. If no payment is involved and it's just a contract they need to sign, then you would use Conditions for the minute changes your clients want.

Related guide: Smart-Forms-Using-Conditional-Logic