Both are durable, alternative metals — but they feel and perform quite differently:
Tungsten is the hardest and heaviest. It's virtually scratch-proof and holds its polish for years, making it ideal for people who want a zero-maintenance ring. The trade-off is that it can crack under extreme impact (like a sharp blow to concrete) and cannot be resized.
Titanium is the lightest of the two — noticeably so. It's hypoallergenic, extremely comfortable for all-day wear, and great for people not used to wearing rings. It scratches more easily than tungsten but is easier to live with for those who find tungsten too heavy.