Electronic payment players PayPal Holdings (PYPL) and rival Square (SQ) are clashing in a critical race to improve digital wallet apps. It's a contest that could have a profound effect on both PayPal stock and Square stock for 2021 and beyond.

The two payment companies are marketing apps that let shoppers get discounts, make installments and buy cryptocurrencies. Both compete in the digital wallet space with technology giants, including Apple (AAPL), Alphabet's (GOOGL) Google and Amazon.com (AMZN), as well as feisty financial technology startups.

But it's the head-to-head battle between PayPal and Square — and their respective fast-growing Venmo and Cash App services — that draws the most interest on Wall Street.