In the first sentence, play is an infinitive, and playing is a gerund.
Both of those are "non-finite" verbs.
It's really hard to easily express the difference in meaning between infinitive and gerund. Even the Wikipedia article can't address it without being complicated, but a clue is here:
Historically, the -ing suffix was attached to a limited number of verbs to form abstract nouns, which were used as the object of verbs such as like. The use was extended in various ways: the suffix became attachable to all verbs ...
... The present-day result of these developments is that the verbs followed by -ing forms tend to fall into semantic classes.
So ... the overall concept of -ing as a gerund is that it tends to come after specific meaning-groups of verbs. The Wikipedia article provides those groups:
- 'LIKE' AND 'DISLIKE' GROUP - adore, appreciate, (cannot|) bear, (not) begrudge, detest, dislike, (cannot) endure, enjoy, hate, like, loathe, love, (not) mind, mind, prefer, relish, resent, (cannot) stand, (cannot) stomach, (not) tolerate, take to
dread, (not) face. fancy, favour, fear, look forward to
- 'CONSIDER' GROUP - anticipate, consider, contemplate, debate, envisage, fantasise, imagine, intend, visualise
- 'REMEMBER' GROUP - forget, miss, recall, recollect, regret, remember, (cannot) remember
- 'RECOMMEND' GROUP - acknowledge, admit, advise, advocate, debate, deny, describe, forbid, mention, prohibit, propose, recommend, report, suggest, urge
- 'INVOLVE' GROUP - allow, entail, involve, justify, mean, necessitate, permit, preclude, prevent, save
- 'POSTPONE' GROUP - defer, delay, postpone, put off
- 'NEED' GROUP - deserve, need, require, want
- 'RISK' GROUP - chance, risk
Using playing instead of play tilts the verb's meaning towards one of those groups. When playing is said, you're possibly elevating it to an event that one can be involved in, or maybe something with some risk attached, or possibly one of the above shades of meaning.